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The topology of representation varieties Bergeron, Maxime Octave 2016

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The Topology of Representation VarietiesbyMaxime Octave BergeronBSc Honours Mathematics, McGill University, 2011MSc Mathematics, McGill University, 2012a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mathematics)The University of British Columbia(Vancouver)August 2016c© Maxime Octave Bergeron, 2016AbstractThe crux of this thesis is the topology of representation varieties. To be more precise, let Gbe a complex reductive linear algebraic group and let K ⊂ G be a maximal compact sub-group. Given a nilpotent group Γ generated by r elements, we consider the representationspaces Hom(Γ, G) and Hom(Γ,K ) with the natural topology induced from an embeddinginto Gr and Kr respectively. Our main result shows that there is a strong deformationretraction of Hom(Γ, G) onto Hom(Γ,K ). We also obtain a strong deformation retrac-tion of the geometric invariant theory quotient Hom(Γ, G)//G onto the ordinary quotientHom(Γ,K )/K. Using these deformations, we then describe the topology of these spaces.iiPrefaceThis thesis is an amalgam of two manuscripts [7, 8] previously published in Geometry &Topology and the Journal of Group Theory. The introduction of this thesis is a modified andexpanded version of the respective introductions of these manuscripts. The mathematicalresults found in [7] are the fruit of independent work by the author under the guidance ofAlexandra Pettet, Lior Silberman and Juan Souto; they make up the second (and main)chapter of this thesis. The content of the third chapter can also be found in [8] and is theresult of joint work with Lior Silberman. Given the nature of this work, there is no clearway to separate the individual contributions to this chapter.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Topology of Nilpotent Representations in Reductive Groups andtheir Maximal Compact Subgroups . . . . . . . . . . . . . . . . . . . . . . 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 Historical remarks and applications . . . . . . . . . . . . . . . . . . . 102.1.2 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 A crash course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Jordan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Nilpotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . 142.2.4 Complex reductive algebraic groups . . . . . . . . . . . . . . . . . . 152.3 Algebraic actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Geometric invariant theory . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 The Kempf–Ness Theorem . . . . . . . . . . . . . . . . . . . . . . . 182.3.3 The Neeman–Schwarz Theorem . . . . . . . . . . . . . . . . . . . . . 192.4 The representation variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.1 The Frobenius norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 21iv2.4.2 The Kempf–Ness set . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 The scaling operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.1 Scaling eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.2 Scaling the Kempf–Ness set . . . . . . . . . . . . . . . . . . . . . . . 262.6 The real and complex cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6.1 Real Kempf–Ness theory . . . . . . . . . . . . . . . . . . . . . . . . . 282.6.2 Proofs of Theorem I and Theorem II . . . . . . . . . . . . . . . . . . 292.7 Expanding nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 A Note on Nilpotent Representations . . . . . . . . . . . . . . . . . . . . . 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 An interesting bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Proofs of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46vAcknowledgmentsI would like to thank Juan Souto for introducing me to the topology of representationvarieties and, in some sense, mathematics as a whole; needless to say, it changed my life.I would also like to thank him, along with Alexandra Pettet and Lior Silberman, for theirguidance in the form of countless interesting, friendly and often fruitful conversations. Inparticular, I thank Lior for allowing me to use some of our joint work in this thesis.I would like to express my gratitude to Alejandro Adem, Kee Lam, Fok-Shuen Leung,Zinovy Reichstein and Dale Rolfsen for their ongoing support in various guises.I would like to thank Man Chuen Cheng, Ben Cohen Wallace, Galo Higuera Rojo, TomHutchcroft, Justin Martel, Tali Pinsky, Vasu Tewari and Bernardo Villarreal Herrera forcarefully listening to my many rambles.I would like to acknowledge the financial assistance I received from UBC in the formof a Four Year Fellowship as well as from the National Sciences and Engineering ResearchCouncil of Canada in the form of an Alexander Graham Bell Canada Graduate Scholarship.Last, but not least, I would like to thank my (ever growing) family for getting me thisfar in life, and then further.(o o)viChapter 1IntroductionThis thesis is motivated by a desire to understand the space of representations of finitelygenerated groups into linear algebraic groups. In the classical case where the source isfinite, this essentially reduces to the correspondence between linear representations andcharacters, due to Frobenius. On the other hand, when the source is infinite, the ana-logue of character theory is the parametrization of representations by geometric varieties.Although these so-called representation varieties are interesting in their own right, theyarise concretely in various settings such as mathematical physics, symplectic geometry andgauge theory. There, one often encounters basic questions about their topology; for in-stance, one might need to count their connected components. In this context, the author’slong-standing goal has been to understand the topology of these spaces by exploiting theiralgebro-geometric structure.To be more precise, let Γ be a group generated by r elements and let G be a complexlinear algebraic group such as SLnC. Since a representation of Γ inG is uniquely determinedby the image of a generating set, the space Hom(Γ, G) of homomorphisms ρ : Γ → G canbe realized as an affine algebraic set, carved out of Gr by the (polynomial) relations of Γ.Indeed, if Γ has a presentation 〈γ1, . . . , γr |R1, R2, R3, . . .〉 with generators γi and relationsRj , then we can identify Hom(Γ, G) with the set of tuples (g1, . . . , gr) ∈ Gr satisfying thegiven relations. It is well known and otherwise easy to see that this geometric structure isindependent of the chosen presentation of Γ.Remark 1. It is interesting to note that, by the Hilbert Basis Theorem [28], there is agroup Γ′ with presentation 〈γ1, . . . , γr |R′1, R′2, . . . , R′m〉 such that the variety Hom(Γ, G)coincides with the variety Hom(Γ′, G). In other words, we may always assume without loss1of generality that our group Γ is finitely presented.Example 1. When Γ = Zr, the space Hom(Zr, G) can be identified with the space{(g1, g2, . . . , gr) ∈ Gr : gigj = gjgi for all i and j} (1.1)of commuting r-tuples in G. The nice thing about this example is that it is alreadycomplicated enough to be interesting yet simple enough to be understood concretely.As a complex variety, Hom(Γ, G) admits a natural Hausdorff topology obtained froman embedding into affine space. If K is a maximal compact subgroup of G, for instanceSUn ⊂ SLnC, we can then endow Hom(Γ,K ) ⊂ Hom(Γ, G) with the subspace topology. Itis much easier to answer topological questions for the compact space Hom(Γ,K ). Indeed,most of what is known about the algebraic topology of representation spaces stems fromthe work of Adem and Cohen [2] and its follow-ups in this compact setting. This comesfrom the fact that many tools from algebraic topology and the theory of Lie groups breakdown when G is not compact. The fascinating thing here is that, in some cases, theseproblems can be circumvented because the entire representation variety retracts onto thespace of representations in a compact subgroup. In order to make this precise, recall thata strong deformation retraction of a space X onto a subspace Y ⊂ X is a continuous mapf : X × [0, 1]→ X (1.2)such that1. f(x, 0) = x for all x ∈ X,2. f(x, 1) ∈ Y for all x ∈ X, and3. f(y, t) = y for all y ∈ Y and t ∈ [0, 1].The main result of this thesis is the following:Theorem I (Bergeron [7]). Let Γ be a finitely generated nilpotent group and let G be thegroup of complex or real points of a (possibly disconnected) reductive linear algebraic group,defined over R in the latter case. If K is any maximal compact subgroup of G, then thereis a (K– equivariant) strong deformation retraction of Hom(Γ, G) onto Hom(Γ,K ). Inparticular, Hom(Γ, G) and Hom(Γ,K ) are homotopy equivalent.2The assumption that G be reductive in Theorem I is necessary. Combining ideas ofMal’cev and Dyer, one can produce a (nilpotent) lattice Γ in a unipotent Lie group U forwhich the space Hom(Γ, U ) is disconnected. Viewing U as an algebraic group and keepingin mind that a maximal compact subgroup K ⊂ U is necessarily trivial, we see that theinclusion Hom(Γ,K ) ↪→ Hom(Γ, U ) is not even a bijection of connected components. Onthe other hand, it goes without saying that Theorem I does not hold for arbitrary finitelygenerated groups Γ. Indeed, if n ≥ 3 and Γ ⊂ SLnR is a cocompact lattice, then it wasshown by Selberg [49] that the component of the tautological representation Γ ↪→ SLnRconsists of the conjugates of Γ and, in particular, is disjoint from Hom(Γ, SOnR). Onceagain the inclusion Hom(Γ,K ) ↪→ Hom(Γ, G) is not a bijection of pi0.It may also be worth mentioning that Theorem I has already found applications beyondthe author’s own work. For example, Higuera [27] made use of it in his PhD Thesis toextend his results on nilpotent representations in SU2 to SL2C, Adem and Cheng [1] usedit to generalize their work on almost commuting matrices in unitary groups to generallinear groups and, in a different direction, it has been used in the development of nilpotentK-theory by Adem, Go´mez, Lind and Tillmann [5].Before going any further, it should be noted that Theorem I had two predecessors.When Γ is an abelian group, it is due to Pettet and Souto [43] and when Γ is an expandingnilpotent group it is is an unpublished result of Silberman and Souto. In fact, our approachwas the fruit of a successful attempt to replace the ad-hoc homotopy-theoretic methodsused by these authors with completely different techniques coming from algebraic geometry,notably, from Geometric Invariant Theory (GIT). In particular, even in the case whereΓ = Zr, the proof is entirely new.Remark 2. As the reader may be aware, there is always a strong deformation retractionof G onto K given by the polar decomposition. As such, they may be wondering wherethe difficulty lies in the aforementioned results. Indeed, if Γ = Fr is a free group of rank r,one immediately obtains from this polar deformation thatHom(Fr, G) = Gr ' Kr = Hom(Fr,K ). (1.3)Nevertheless, when Γ is not a free group, there is no guarantee that such a deformation ofG onto K preserves the necessary relations. Indeed, the predecessors to Theorem I alreadyhave rather delicate proofs because of a result by Souto [53] showing that, for n ≥ 8, thereis no retraction of SLnC onto SUn which preserves commutativity.3Let us recall at this point that G acts by conjugation on (the image of) representa-tions and that, when G = SLnC or G = GLnC, representations are in the same G-orbitwhenever their images differ by a change of basis. As such, one often wishes to classifyrepresentations modulo this equivalence relation. Unfortunately, since G is not compact,the naive topological quotient Hom(Γ, G)/G may be in need of repair: the failure of someorbits to be closed prevents it from being Hausdorff. Consider, for instance, the action ofSL2C on Hom(Z,SL2C) = SL2C by conjugation. In this setting it is easy to see that thefollowing orbit is not closed(λ 00 λ−1)(1 10 1)(λ 00 λ−1)−1λ→0−−−→(1 00 1). (1.4)This is why we consider the GIT quotient Hom(Γ, G)//G obtained from the ring of invariantfunctions instead. In purely topological terms, this so-called character variety may beconstructed as the universal quotient in the category of Hausdorff spaces. One of theadvantages of the algebraic techniques used to prove Theorem I is that they also yield ananalogous deformation result for character varieties. This is the second main result of thisthesis:Theorem II (Bergeron [7]). Let Γ be a finitely generated nilpotent group and let G bethe group of complex or real points of a (possibly disconnected) reductive linear algebraicgroup, defined over R in the latter case. If K ⊂ G is any maximal compact subgroup,then there is a strong deformation retraction of the character variety Hom(Γ, G)//G ontoHom(Γ,K )/K.When Γ is abelian, Theorem II is a recent result of Florentino and Lawton [24] (follow-ing up on their previous work for free groups in [23]) and we refer the reader to their paperfor various applications. The main tools from geometric invariant theory used to proveTheorem I and Theorem II are the Kempf–Ness Theorem and the Neeman–Schwarz Theo-rem. More precisely, for Γ and G as in Theorem I, we embed Hom(Γ, G) into the complexvector space ⊕rj=1MnC equipped with the Frobenius norm. Following Richardson, we thendenote byM the set of representations that are of minimal norm within their G–orbit, theso-called Kempf–Ness set of Hom(Γ, G). Our main technical result characterizes this setwhen Γ is nilpotent:Theorem III (Bergeron [7]). Let G ⊂ SLnC be a complex reductive linear algebraic4group and suppose that K = G ∩ SUn ⊂ G is a maximal compact subgroup. If Γ is afinitely generated nilpotent group and M denotes the Kempf–Ness set of the G–subvarietyHom(Γ, G) ⊂ ⊕rj=1MnC equipped with the Frobenius norm, thenM = {ρ ∈ Hom(Γ, G) : ρ(Γ) consists of normal matrices}.Here, it is worth noting that the nilpotency assumption on Γ is necessary; indeed,we shall see that Theorem III is already false for the solvable group Z o Z/4Z. Sincethe Neeman–Schwarz Theorem provides us with a retraction of Hom(Γ, G) onto M, inorder to deduce Theorem I from Theorem III it remains to deform representations ofnormal matrices into representations of unitary matrices. This is accomplished by scalingeigenvalues. Finally, Theorem II is a consequence of Theorem III after applying a corollaryof the Kempf–Ness Theorem.The main motivation behind Theorem I and Theorem II is to understand the topologyof representation spaces. This can now be effectively achieved in many ways. For instance,building on previous work of Pettet, Silberman and Souto, Lior Silberman and the authorhave used them to get a handle on various invariants. To be more precise, recall that anon-abelian finitely generated group Γ is s-step nilpotent if its lower central series, definedinductively byΓ(1) = Γ and Γ(i+1) = [Γ,Γ(i)], (1.5)has Γ(s) non-trivial but Γ(s+1) = {e}. Notice that the epimorphism Γ→ Γ/Γ(i) induces anembeddingHom(Γ/Γ(i), G)→ Hom(Γ, G) (1.6)which, for general groups Γ and G, is not even an open map. Nevertheless, using TheoremI, we will see that the following holds:Theorem IV (Bergeron-Silberman [8]). Let Γ be a finitely generated nilpotent group andlet G be the group of complex points of a (possibly disconnected) reductive linear algebraicgroup. For all i ≥ 2, the inclusionHom(Γ/Γ(i), G)ι−→ Hom(Γ, G)is a homotopy equivalence onto the union of those components of the target intersecting theimage of ι.5Let us now consider Hom(Γ, G) as a pointed space by taking the trivial representationas base point. From this point of view, Theorem IV implies that the connected componentsHom(Γ, G)1 ⊂ Hom(Γ, G) and Hom(Γ/Γ(i), G)1 ⊂ Hom(Γ/Γ(i), G) (1.7)of the trivial representation are homotopy equivalent for all i ≥ 2. This allows us todescribe the homotopy type of the component of the trivial representation in terms ofabelian representations:Hom(Γ, G)1 ' Hom(ZrankH1(Γ;Z), G)1. (1.8)Applying Theorem I to reduce things to the compact case, we can then use the work of Baird[6], Go´mez–Pettet–Souto [25] and Biswas–Lawton–Ramras [11] in this setting to computeits fundamental group and describe its cohomology as well as that of the correspondingcomponent in the character variety.While the results above indicate many similarities between representation spaces ofabelian and non-abelian nilpotent groups, the latter have a much richer topology than theformer. For example, recall that Hom(Zr,SLnC), Hom(Zr, Sp2nC) and the correspondingcharacter varieties are connected for all values of r and n. Using our results, we are ableto show that the situation for non-abelian nilpotent groups is completely different:Theorem V (Bergeron-Silberman [8]). Let G be the group of complex points of a reductivelinear algebraic group. If Γ is a finitely generated nilpotent group which surjects onto afinite non-abelian subgroup of G, then Hom(Γ, G) and Hom(Γ, G)G are both disconnectedtopological spaces.One way to see Theorem V in action is to observe that non-abelian free nilpotent groupsand discrete Heisenberg groups surject onto the non-abelian nilpotent group of order 8.Contrary to popular beliefs, this implies that their representation varieties are essentiallyalways disconnected.Example 2. Consider the discrete Heisenberg groupH3 :=1 x z0 1 y0 0 1 : x, y, z ∈ Z ∼= 〈x, y, z | [x, z] = [y, z] = e, [x, y] = z〉. (1.9)6The phenomenon mentioned above already occurs when we consider representations of H3in SU2. There, Hom(H3,SU2) breaks up into two components. A simple computation showsthat any representation ρ : H3 → SU2 which doesn’t factor through the abelianizationH3/[H3, H3] ∼= Z2 is conjugate to the quaternionic representationx 7→(i 00 −i), y 7→(0 1−1 0)and z 7→(−1 00 −1). (1.10)Since the stabilizer of such a non-abelian representation corresponds to {± Id} ⊂ SU2, wesee that the component of Hom(H3, SU2) consisting of these representations is homeomor-phic to SU2 / ± Id ' RP3. The other component of Hom(H3, SU2) consisting of abelianrepresentations is homeomorphic to Hom(Z2, SU2).Remark 3. The statements of Theorem IV and Theorem V remain true when G is replacedby a compact Lie group K. In fact, we prove them in this setting before obtaining thecomplex reductive case by applications of Theorem I and Theorem II.We conclude this introduction with a brief outline of the thesis. The second chapterconcerns the details behind Theorem I and Theorem II. In the third chapter, we proveTheorem III, Theorem IV and their many corrolaries. Finally, in the concluding chapter,we explore the meaning of these results within the field as a whole and describe our ongoingwork on the many questions that arose along the way.7Chapter 2The Topology of NilpotentRepresentations in ReductiveGroups and their MaximalCompact SubgroupsThis chapter is a transcript of [7] consisting of independent work by the author.2.1 IntroductionLet Γ be a group generated by r elements and let G be a complex linear algebraic group.Since a representation of Γ in G is uniquely determined by the image of a generatingset, the space Hom(Γ, G) of homomorphisms ρ : Γ → G can be realized as an affinealgebraic set carved out of Gr by the relations of Γ. It is well known and otherwiseeasy to see that this geometric structure is independent of the chosen presentation of Γ.As a complex variety, Hom(Γ, G) admits a natural Hausdorff topology obtained from anembedding into affine space. If K is a maximal compact subgroup of G, we can then endowHom(Γ,K ) ⊂ Hom(Γ, G) with the subspace Hausdorff topology. Although in general thesetopological spaces may be quite different, in this thesis we show that when Γ is nilpotentand G is reductive they are actually homotopy equivalent:Theorem I. Let Γ be a finitely generated nilpotent group and let G be the group of complexor real points of a (possibly disconnected) reductive linear algebraic group, defined over R8in the latter case. If K is any maximal compact subgroup of G, then there is a (K-equivariant) strong deformation retraction of Hom(Γ, G) onto Hom(Γ,K ). In particular,Hom(Γ, G) and Hom(Γ,K ) are homotopy equivalent.The assumption that G be reductive in Theorem I is necessary. Combining ideas ofMal’cev and Dyer, one can produce a (nilpotent) lattice Γ in a unipotent Lie group Ufor which the space Hom(Γ, U ) is disconnected. Viewing U as an algebraic group andkeeping in mind that a maximal compact subgroup K ⊂ U is necessarily trivial, we seethat the inclusion Hom(Γ,K ) ↪→ Hom(Γ, U ) is not even a bijection of pi0. A completediscussion of this example will be given in the Section 2.7. Similarly, Theorem I does nothold for arbitrary Γ: if n ≥ 3 and Γ ⊂ SLnR is a cocompact lattice, then it was shownby Selberg [49] that the component of the tautological representation Γ ↪→ SLnR consistsof the conjugates of Γ and, in particular, is disjoint from Hom(Γ,SOnR). Once again theinclusion Hom(Γ,K ) ↪→ Hom(Γ, G) is not a bijection of pi0.There is also an interesting contrast between the real and complex cases of TheoremI. For instance, while nilpotent subgroups of O+n,1R (the group of isometries of hyper-bolic n-space) are all virtually abelian, this is far from being true in GLnR. Nevertheless,OnR is a maximal compact subgroup of both GLnR and O+n,1R so Theorem I shows thatHom(Γ,O+n,1R) and Hom(Γ,GLnR) are homotopy equivalent. This kind of example doesnot occur in the complex case because complex reductive linear algebraic groups are de-termined by their maximal compact subgroups.It should be mentioned at this point that Theorem I has two predecessors. When Γ isan abelian group, it is due to Pettet and Souto [43] and when Γ is an expanding nilpotentgroup (c.f. the Section 2.7) it is is due to Lior Silberman and Juan Souto. While theseauthors used homotopy-theoretic methods, in this thesis we rely on algebraic geometryinstead. In particular, even in the case where Γ = Zr our proof is completely new. Oneof the advantages of this approach, coming from geometric invariant theory, is that thecorresponding result for the character variety Hom(Γ, G)//G follows from the proof ofTheorem I:Theorem II. Let Γ be a finitely generated nilpotent group and let G be the group ofcomplex or real points of a (possibly disconnected) reductive linear algebraic group, definedover R in the latter case. If K ⊂ G is any maximal compact subgroup, then there is astrong deformation retraction of the character variety Hom(Γ, G)//G onto Hom(Γ,K )/K.When Γ is abelian, Theorem II is a recent result of Florentino and Lawton [24] (follow-9ing up on their previous work for free groups in [23]) and we refer the reader to their paperfor various applications. The main tools from geometric invariant theory used to proveTheorem I and Theorem II are the Kempf-Ness Theorem and the Neeman-Schwarz Theo-rem. More precisely, for Γ and G as in Theorem I, we embed Hom(Γ, G) into the complexvector space ⊕rj=1MnC equipped with the Frobenius norm. Following Richardson, we thendenote byM the set of representations that are of minimal norm within their G-orbit, theso-called Kempf-Ness set of Hom(Γ, G). Our main technical result characterizes this setwhen Γ is nilpotent:Theorem III. Let G ⊂ SLnC be a complex reductive linear algebraic group and supposethat K = G∩ SUn ⊂ G is a maximal compact subgroup. If Γ is a finitely generated nilpotentgroup and M denotes the Kempf-Ness set of the G-subvariety Hom(Γ, G) ⊂ ⊕rj=1MnCequipped with the Frobenius norm, thenM = {ρ ∈ Hom(Γ, G) : ρ(Γ) consists of normal matrices}.Here, it is worth noting that the nilpotency assumption on Γ is necessary; in Section2.4.2 we show that Theorem III is false for the solvable group ZoZ/4Z. Since the Neeman-Schwarz Theorem provides us with a retraction of Hom(Γ, G) onto M, in order to deduceTheorem I from Theorem III it remains to deform representations of normal matrices intorepresentations of unitary matrices. This is accomplished by scaling eigenvalues. Finally,Theorem II is a consequence of Theorem III after applying a corollary of the Kempf-NessTheorem.2.1.1 Historical remarks and applicationsIn coarsest terms, this thesis is concerned with the geometric classification of represen-tations of finitely generated groups. Classically, this subject finds its roots in Poincare´’swork on monodromy groups of linear homogeneous equations and geometric invariant the-ory. More recently, the study of representation varieties has had impacts in a variety ofcontexts and we mention here but a few.From a first point of view, the geometry of representation varieties can be used to deducealgebraic information about the representation theory of a group. For instance, Lubotzkyand Magid [35] obtained such information for the GLnC representations of a finitely gen-erated group Γ by using Weil’s results on the Zariski tangent space of Hom(Γ,GLnC).10Incidentally, these techniques were most effective when Γ was assumed to be nilpotent,partly because this ensured a special vanishing property of its first group cohomology.One can also study representation varieties from a purely differential geometric pointof view. Here, when Γ is the fundamental group of a smooth manifold M , Hom(Γ,K ) canbe identified with the space of pointed flat connections on principal K-bundles over M .These spaces lie at the intersection of various fields including gauge theory and symplecticgeometry as illustrated for surfaces in Jeffrey’s survey [30]. More precisely, there is a naturalconjugation action of K on Hom(Γ,K ) and the quotient Hom(Γ,K )/K corresponds to themoduli space of flat-connections on principal K-bundles over M . In the case where M iscompact and Ka¨hler, the geometric invariant theory quotient Hom(Γ, G)//G correspondsto the moduli space of polystable G-bundles over M via the Narasimhan-Seshadri Theorem(see Narasimhan-Seshadri [40] and Simpson [52]). More generally, these character varietieshave attracted a lot of attention lately as shown in Sikora’s survey [51, Section 11] and thereferences therein.More topologically, the work in supersymmetric Yang-Mills theory and mirror sym-metry of Witten in [55] and [56] sparked an interest in the connected components of freeabelian representation varieties as seen in Kac-Smilga [31] and Borel-Friedman-Morgan[13]. Here, there has been much recent development in the study of higher topologicalinvariants stemming from work of A´dem and Cohen [2] in the compact case. Prior toPettet and Souto’s work in [43], most known topological invariants concerned spaces ofrepresentations into compact groups; it was their deformation retraction which allowedmany of these results to be extended to representations into reductive groups. We brieflyindicate how this strategy works in the nilpotent case, referring the reader to [43] for moreapplications of this type.Suppose for the sake of concreteness that Γ is a torsion free nilpotent group and G is acomplex reductive linear algebraic group with a given maximal compact subgroup K ⊂ G.Denote by Hom(Γ,K )1 the connected component of Hom(Γ,K ) containing the trivial rep-resentations. In [25], Go´mez, Pettet and Souto compute that pi1( Hom(Zr,K )1) ∼= pi1(K)rand use the retraction constructed in [43] to conclude that pi1( Hom(Zr, G)1) ∼= pi1(G)r. Itis not hard to see in our case that Hom(Γ,K )1 coincides with the representations factor-ing through the abelianization of Γ lying in the component of the trivial representation.These observations can be combined with Theorem I to compute that pi1( Hom(Γ, G)1) ∼=pi1(G)rankH1(Γ;Z). The topology of Hom(Γ,K ) will be analyzed in greater detail in the thirdchapter.112.1.2 Outline of the chapterWe begin in Section 2.2 by establishing some notation and refreshing the reader’s memorywith some basic facts about algebraic groups. Then, in Section 2.3, we introduce the maintechnical tools from Kempf-Ness theory that we shall need. After these preliminaries, webegin working with representation varieties in Section 2.4 where we prove Theorem III.Then, in Section 2.5, we show how eigenvalues can be scaled to complete the proof ofTheorem I in the complex case. Finally, in Section 2.6, we show how the proof carries overto the real case and we prove Theorem II.2.2 Algebraic groupsIn this preliminary section, we refresh the reader’s memory with some basic facts aboutlinear algebraic groups, referring to Borel’s book [12] for the details. In his own words:According to one’s taste about naturality and algebraic geometry, it is possible to give severaldefinitions of linear algebraic groups. Here, we will favour a concrete and slightly pedestrianapproach since it is all that we shall need. There is only one possibly nonstandard result,Lemma 2.2.4.2.2.1 A crash courseAn affine algebraic group is an affine variety endowed with a group law for which thegroup operations are morphisms of varieties, i.e., polynomial maps. For our purposes, analgebraic group G shall mean the group of complex (or real) points of an affine algebraicgroup (defined over R in the latter case) and an affine variety shall mean an affine algebraicset, i.e., the zero locus of a family of complex polynomials. It turns out that affine algebraicgroups are linear so we will always identify them with a Zariski closed subgroup of SLnC.Remark 4. We shall have the occasion to consider two different topologies on varieties andtheir subsets, the classical Hausdorff topology and the Zariski topology. Unless we specifyotherwise, all references to topological concepts will refer to the Hausdorff topology. Forinstance, a closed set is closed in the Hausdorff topology and a Zariski closed set is closedin the Zariski topology. Nevertheless, since a complex algebraic group is connected in theHausdorff topology if and only if it is connected in the Zariski topology, we shall make nosuch distinction when referring to their identity component.Let G ⊂ SLnC be an algebraic group.121. We say that a subgroup L is an algebraic subgroup of G if it is Zariski closed. Sincean arbitrary subgroup H of G isn’t necessarily algebraic, we often pass from H to itsZariski closure A (H). It turns out that A (H) is also a subgroup of G and, hence,the smallest algebraic subgroup of G containing H.2. We say that G is connected if it is connected in the Zariski topology. Much of thebehaviour of G is governed by the connected component of its identity element whichwe denote by G◦. This is always a normal algebraic subgroup of finite index in G.The following Theorem due to Mostow [38] indicates one of the many roles played byG◦; it shall be used several times in this thesis.Theorem 2.2.1 (Conjugacy Theorem). An algebraic group G contains a maximalcompact subgroup and all such subgroups are conjugate by elements of G◦.3. Lastly, since G is a smooth variety, it has a well defined tangent space at the identitythat we denote by g. The Lie algebra of G is the set g endowed with the usual Liealgebra structure given by the bracket operation on derivations.2.2.2 Jordan decompositionLet V be a finite dimensional complex vector space and recall the following elementarydefinitions from linear algebra; an endomorphism σ of V is said to be:1. Nilpotent if σn = 0 for some n ∈ N.2. Unipotent if σ − Id is nilpotent.3. Semisimple if V is spanned by eigenvectors of σ.If G ⊂ SLnC is an algebraic group, we say that g ∈ G is semisimple (resp. unipotent)if it is semisimple (resp. unipotent) as an endomorphism of Cn. One can show that thisdefinition does not depend on the chosen embedding of G in SLnC. In fact, we have thefollowing useful theorem:Jordan Decomposition Theorem. Let G be a linear algebraic group.1. If g ∈ G then there are unique elements gs and gu in G so that gs is semisimple, guis unipotent and g = gsgu = gugs.132. If ϕ : G → G′ is a morphism of algebraic groups, then ϕ(g)s = ϕ(gs) and ϕ(g)u =ϕ(gu).We denote the set of all unipotent (resp. semisimple) elements of G by Gu (resp. Gs).Although the set Gs is seldom Zariski closed, Gu is always a Zariski closed subset of Gcontained in G◦. In fact, we say that an algebraic group G is unipotent if G = Gu. We cannow define the class of algebraic groups that we will be most concerned with in this thesis:Definition 2.2.2. The unipotent radical of an algebraic group G is its maximal connectednormal unipotent subgroup. An algebraic group G is said to be reductive if its unipotentradical is trivial.Example 3. The classical groups are all reductive, e.g., SLnC, GLnC and SpnC.2.2.3 Nilpotent algebraic groupsLet A and B be any subgroups of G and denote by [A,B] the commutator subgroup ofG abstractly generated by elements of the form aba−1b−1, a ∈ A and b ∈ B. The lowercentral series of G is defined inductively by the ruleG =: G(0) D [G,G(0)] =: G(1) D [G,G(1)] =: G(2) D . . .D [G,G(n)] =: G(n+1) D . . .and one says that G is nilpotent if for some n ≥ 0 we have G(n) = {e}. This definitionmakes sense in the context of algebraic groups since all of the groups in the lower centralseries are algebraic, i.e., Zariski closed. In fact, we have the following classification resultwhich may be found in Borel [12, III.10.6]:Proposition 2.2.3. Let G be an algebraic group and recall that we denote its subset ofsemisimple (resp. unipotent) elements by Gs (resp. Gu). If G is connected and nilpotent,then Gs and Gu are algebraic subgroups of G and G ∼= Gs ×Gu.We end this subsection with a structural result that will be useful later on. Althoughit may be known to the experts, we provide a proof since we were unable to find it in theliterature.Lemma 2.2.4. If N is a (possibly disconnected) nilpotent reductive algebraic group, then1. N consists of semisimple elements.142. N has a unique maximal compact subgroup.3. The connected component N◦ is contained in the centre of N .The following argument was shown to us by Lior Silberman.Proof. Recall from Proposition 2.2.3 that the connected component of a nilpotent algebraicgroup admits a direct product decomposition N◦ = N◦s × N◦u . Since N◦ is reductive, N◦umust be trivial and N◦ = N◦s consists of semisimple elements. The proof of claim (1) isthen completed by observing that Nu ⊂ N◦. Moreover, assuming for the moment that N◦is central in N , claim (2) follows at once from the Conjugacy Theorem 2.2.1.It remains to prove that N◦ is central in N . To do so, it suffices to show that the adjointaction [12, Chapter I.3.13] of N on its Lie algebra n is trivial. Seeking a contradiction,suppose that Ad(g) acts nontrivially on n for some g ∈ N . Since g is semisimple, theJordan Decomposition Theorem ensures that Ad(g) is semisimple and must therefore havean eigenvalue λ 6= 1. If X ∈ n is an eigenvector of Ad(g) corresponding to λ, we have that[g, exp(tX)] = getXg−1e−tX = eAd(g)(tX)−tX = et(λ−1)X .As such, the iterated commutators [g, [g, [g, . . . , [g, [g, eX ]] . . .]]] are all non-trivial and thelower central series of N does not terminate in finitely many steps. This is a contradiction.2.2.4 Complex reductive algebraic groupsOne of the advantages of working over C is the strong relation between reductive groupsand their maximal compact subgroups. We illustrate this by following the treatment inSchwarz [48].Given a compact Lie group K, the Peter-Weyl Theorem provides a faithful embeddingK ↪→ GLnR for some n. Identifying K with its image realizes it as a real algebraic subgroupof GLnR. We then define the complexification G := KC to be the vanishing locus in GLnCof the ideal defining K. The group G is a complex algebraic group which is independentup to isomorphism of the embedding provided by the Peter-Weyl Theorem. The followingproperties of complexifications of compact Lie groups are well known (see, for instance,Onishchik and Vinberg [42, Chapter 5]):151. The Lie algebra g of G has a natural splitting in terms of the Lie algebra k of K:g = k⊕ ik.2. The group K is a maximal compact subgroup of G. Moreover, it is Zariski dense inG, i.e., A (K) = G.3. G = K · exp(ik) ∼= K × exp(ik) where ∼= is a diffeomorphism. (Contracting exp(ik) isone way to see that K ↪→ G is a homotopy equivalence.)Example 4. Viewing SUn as a compact Lie group, we can realize it as a real algebraicsubgroup of GL2nR. In this case, the complexification of SUn is isomorphic to SLnC ⊂GL2nC.Remark 5. The decomposition G = K ·exp(ik) is often called the polar decomposition.Indeed, for any linear realization G ⊂ SLnC, it is a Theorem of Mostow [38] that for somem ∈ SLnC the corresponding polar decomposition of mGm−1 ⊂ SLnC coincides with theusual polar decomposition in SLnC. We shall use this decomposition several times in theproof of our main theorem.As one might suspect, the following theorem holds [42, Chapter 5, Section 2]:Theorem 2.2.5. A complex linear algebraic group is reductive if and only if it is thecomplexification of a compact Lie group.2.3 Algebraic actionsLet V be a complex vector space and suppose that the complex reductive algebraic groupG is contained in GL(V ). The goal of this preliminary section is to study the algebraicand topological structures of a variety X ⊂ V that is stable under the action of G ⊂GL(V ). This will lead us to the main known result used in our proof: the Neeman–SchwarzTheorem.In keeping with the general spirit of this thesis, recall that we favour a concrete approachto algebraic groups. As such, we also adopt the following concrete point of view for algebraicactions. A G–variety is a variety X ⊂ V acted upon algebraically by G ⊂ GL(V ). Recallthat the orbit of x ∈ X, denoted G · x, is the set of all g · x with g ∈ G. A G–subvariety ofX is then a G–variety Y ⊂ X ⊂ V such that G · y ⊂ Y for every y ∈ Y . If X and Y areG–varieties, a morphism α : X → Y is G– equivariant if α(g · x) = g · α(x).16Remark 6. There is no real loss of generality in our concrete approach to G–varieties.Indeed, any affine variety endowed with a reductive algebraic group action is equivariantlyisomorphic to a closed G–subvariety of a finite dimensional complex vector space (see, forinstance, Brion [15, Proposition 1.9]).2.3.1 Geometric invariant theoryOnce a variety has been equipped with an algebraic group action, one might be temptedto take quotients. Unfortunately, a naive construction of the quotient fails to produce avariety because the orbits of points are not always closed. Informally, geometric invarianttheory remedies this situation by altering the quotient and gluing “bad orbits” together.What follows is intended to be a brief reminder of this construction; the reader unfamiliarwith these concepts is invited to consult Brion [15] for the details.Let X be a G–variety and recall that this endows the ring of regular functions C[X]with a natural action of G. One would expect that the ring of regular functions of thequotient of X by G should correspond to those functions in C[X] which are invariantunder G. Indeed, we define the affine Geometric Invariant Theory (GIT) quotient as thecorresponding affine variety which we denote byX//G := Specm(C[X]G).The topological space underlying X//G has a nice description of independent interest.The key observation here is that the closure of every G– orbit contains a unique closedorbit. Consequently, points of X//G correspond bijectively with the closed G– orbits. Infact, if we denote the set of closed G– orbits by X//topG and consider the quotient mappi : X → X//topG sending x ∈ X to the unique closed orbit in G · x, then X//topG (equippedwith the quotient topology) is homeomorphic to X//G.Remark 7. This so-called topological Hilbert quotient X//topG was investigated by Lunain [36] and [37] to understand the orbit space of real and complex algebraic group actions.In particular, he showed that X//topG is the universal quotient in the category of Hausdorffspaces and in the category of complex analytic varieties.Definition 2.3.1. Let X be a G–variety. A point x ∈ X is said to be polystable if itsorbit G · x ⊂ X is Zariski closed. Since G · x is Zariski open in its Zariski closure, this is17equivalent to requiring that G · x be closed in the Hausdorff topology. We denote the setof polystable points of X by Xps.Example 5. Let us consider the simple case whereG acts on itself by inner automorphisms.Since our definition of a G–variety requires an ambient vector space, we think of G as aG–subvariety of the complex vector space of n × n matrices G ⊂ MnC where G acts onMnC by conjugation. In this setting, it is well known that x ∈ G is polystable if and onlyif x is semisimple.Similarly, we can consider the diagonal action of G on Gr given by g · (x1, . . . , xr) :=(gx1g−1, . . . , gxrg−1). Here, Richardson [46, Theorem 3.6] generalized the previous charac-terization of polystability. The following theorem will play an important part in the proofof Theorem III in Section 2.4.2.Richardson’s Theorem. Let G be a complex reductive linear algebraic group. For everytuple (x1, . . . , xn) ∈ Gn, denote by A (x1, . . . , xn) the algebraic subgroup of G generated bythe elements {x1, . . . , xn}. If we consider the diagonal action of G on Gn by inner auto-morphisms, then (x1, . . . , xn) ∈ Gn is polystable if and only if A (x1, . . . , xn) is reductive.2.3.2 The Kempf–Ness TheoremLet us now consider the entire complex vector space V as a G–variety via the algebraicinclusion G ⊂ GL(V ). If K ⊂ G is a maximal compact subgroup, there is no loss ofgenerality in assuming that V is equipped with a K– invariant Hermitian inner product.It turns out that the associated norm || · || on V sheds a lot of light on its polystablepoints. Indeed, one can determine if a vector v ∈ V is polystable by understanding howits length changes as we move along its orbit G · v. This variation in norm is captured bythe Kempf–Ness function which is defined for every v ∈ V by the ruleΨv : G→ R , g 7→ ||g · v||2. (2.1)Kempf and Ness [32, Theorem 0.2-0.3] used the convexity of these functions to formulatea simple characterization of polystability:Kempf–Ness Theorem. Let G ⊂ GL(V ) and suppose that V is endowed with a K-invariant norm as above. If X ⊂ V is a G–subvariety, then:1. The point x ∈ X is polystable if and only if Ψx has a critical point.182. All critical points of Ψx are minima.Moreover, if Ψx(e) is a minimum then K · x = {y ∈ G · x : ||y|| = ||x||}.With this theorem in mind, we can state the most important definition of this thesis.As above, let G ⊂ GL(V ) where V is endowed with a K-invariant norm. Then, theKempf–Ness set of V is defined as the Minimal vectorsMV := {v ∈ V : ||g · v|| ≥ ||v|| for all g ∈ G}. (2.2)Note that the Kempf–Ness Theorem ensures that MV ⊂ V ps and, although V ps is anintrinsically defined subset of V , MV depends on the choice of the invariant norm on V .If X ⊂ V is a G–subvariety, we define the Kempf–Ness set of X as MX :=MV ∩X.The primary reason for calling the collection of minimal vectors the “Kempf–Ness set”is the following corollary [48, Corollary 4.7] of the Kempf–Ness Theorem. It allows GITquotients to be topologically represented via ordinary quotients and will play a roˆle in theproof of Theorem II.Corollary 2.3.2. The geometric invariant theory quotient X//G is homeomorphic (in theHausdorff topology) to the ordinary quotient MX/K.Remark 8. Whenever the Kempf–Ness set under consideration is clear from the contextwe shall omit the subscript and refer to it simply as M.2.3.3 The Neeman–Schwarz TheoremAs above, let G ⊂ GL(V ) act on the complex vector space V and suppose the latter isendowed with a K– invariant norm || · ||. The following theorem [48, Theorem 5.1] is thekey known result used in the proof of our main theorems.Neeman–Schwarz Theorem. Let the complex vector space V be a G–variety endowedwith a K– invariant norm. If M denotes the Kempf–Ness set of V , then there is a K–equivariant strong deformation retraction of V onto M which preserves G–orbits.More precisely, there is a map ϕ : V × [0, 1]→ V with the following properties:1. ϕ0 = IdV , ϕt|M = IdM, ϕ1(V ) =M2. ϕt(v) ∈ G · v for 0 ≤ t < 1193. ϕ1(v) ∈ G · v.For our purposes, the following corollary shall be most important:Corollary 2.3.3. Let X be a G–subvariety of V and recall thatMX :=M∩X. Restrictingϕ, we obtain a K– equivariant strong deformation retraction of X onto MX .Remark 9. Initially, Neeman [41, Theorem 2.1] had proved that there was a strong de-formation retraction of V onto a neighbourhood ofM. To do so, following a suggestion ofMumford, he considered the negative gradient flow of a function assigning to each v ∈ Vthe “size” of the differential (dΨv)e. However, he believed that the local flow could beextended to a retraction of V onto M and conjectured a functional inequality that wouldallow him to do so. Later, he realized that this was in fact a special case of an inequalitydue to  Lojasiewicz [34]. Finally a complete account of the proof of the Neeman–SchwarzTheorem was given by Schwarz in [48, Theorem 5.1].Remark 10. Although we have been working over C, Kempf–Ness theory was extended tothe real setting by Richardson and Slodowy. For example, in the case of Richardson’s The-orem it is shown in Richardson [46, Theorem 11.4] that (x1, . . . , xn) ∈ G(R)n is polystablefor the action of G(R) if and only if (x1, . . . , xn) ∈ Gn is polystable for the action of G. Forthe Kempf–Ness Theorem, its Corollary and the Neeman–Schwarz Theorem, it is shown inRichardson–Slodowy [47] that the same results hold when one replaces G by its group ofreal points G(R). This will be elaborated upon in Section 2.6 when we prove the real caseof Theorem I.2.4 The representation varietyLet Γ be a finitely generated group and let G be a complex reductive linear algebraicgroup. In a coarse sense, this thesis is concerned with the finite dimensional representationtheory of Γ in G. When Γ is finite, this essentially reduces to the correspondence betweenlinear representations and characters due to Frobenius. When Γ is infinite, the analogue ofcharacter theory is the parametrization of representations by geometric varieties. In thissection, we study the topology of such varieties.In order to endow the space Hom(Γ, G) of homomorphisms ρ : Γ→ G with a geometricstructure, it is convenient to work with a fixed generating set γ1, . . . , γr of Γ. Since anyrepresentation ρ : Γ → G is uniquely determined by the image ρ(γi) of its generators, the20evaluation mapHom(Γ, G)→ G× . . .×G = Gr, ρ 7→ (ρ(γ1), . . . , ρ(γr)) (2.3)is a bijection between Hom(Γ, G) and a Zariski closed subset of Gr carved out by therelations of Γ. It is well known (see, for instance, Lubotzky and Magid [35]) and otherwiseeasy to see that the geometric structure induced on Hom(Γ, G) by this bijection doesn’tdepend on the chosen presentation of Γ.Remark 11. With the hope that no confusion shall arise, we denote elements of Hom(Γ, G)by ρ when we consider them as homomorphisms and by tuples (m1, . . . ,mr) when weidentify them with points in Gr.The action of G on itself by conjugation induces a diagonal action of G on the varietyHom(Γ, G) ⊂ Gr. Explicitly, this action is given by the ruleG×Hom(Γ, G)→ Hom(Γ, G), g · (m1, . . . ,mr) := (gm1g−1, . . . , gmrg−1).Since two representations are in the same G–orbit when their image differs by a change ofbasis, one is often interested in classifying them modulo the action of G. This can be doneby considering the geometric invariant theory quotient Hom(Γ, G)//G which happens toparametrize isomorphism classes of completely reducible representations. Describing thesevarieties geometrically might be viewed as analogous to determining the characters of afinite group.As an affine variety, Hom(Γ, G) also inherits a natural Hausdorff topology given from anembedding into affine space. In fact, it coincides with the subspace topology of Hom(Γ, G) ⊂Gr when G is viewed as a Lie group. If K ⊂ G is any subgroup, there is a natural inclusionHom(Γ,K ) ↪→ Hom(Γ, G) so we endow Hom(Γ,K ) with the subspace topology. Theorem Istates that under certain assumptions there is a deformation retraction of Hom(Γ, G) ontoHom(Γ,K ). The goal of this section is to show how the algebro-geometric tools developedin the previous section can be used in this setting.2.4.1 The Frobenius normLet G be a (possibly disconnected) complex reductive linear algebraic group and let K ⊂ Gbe a maximal compact subgroup. In this subsection, we complete an intermediate step in21the proof of Theorem I by showing how the Neeman–Schwarz Theorem can be applied toHom(Γ, G). Keeping with the general spirit of this thesis, instead of working abstractlywith our algebraic group G, we choose a matrix representation G ↪→ SLnC for whichK = G ∩ SUn.In order to apply the algebro-geometric results from Section 2.3, we need to endowHom(Γ, G) with the structure of an affine G–variety. To do this, consider G ⊂ SLnC as avariety in the vector space MnC of n×n complex matrices. Using the natural identificationsarising from the evaluation map (2.3), we can realize Hom(Γ, G) as an embedded varietyHom(Γ, G) ⊂ Gr ⊂ ⊕rj=1MnC.Once this has been done, the diagonal action of G ⊂ SLnC on ⊕rj=1MnC by simultaneousconjugation endows Hom(Γ, G) with the structure of a G–subvariety of the vector space⊕rj=1MnC ∼= Crn2.In this setting, there is a prototypical K– invariant norm on ⊕rj=1MnC. Recall that thevector space MnC admits a natural Hermitian inner product called the Frobenius innerproduct (see, for instance, Horn and Johnson [29, Chapter 5]). If we denote the adjoint ofa matrix a by a∗ := a t, then this inner product is defined by 〈a, b〉 := trace(a∗b) for a, b ∈MnC. It is easy to check that for any unitary matrix u ∈ Un we have 〈uau−1, ubu−1〉 =〈a, b〉. The Frobenius inner product can then be extended “coordinate wise” to ⊕rj=1MnCby the rule〈(a1, . . . , ar), (b1, . . . , br)〉 :=r∑j=1〈aj , bj〉and the associated norm on ⊕rj=1MnC is unitary invariant (and therefore G ∩ SUn = K–invariant).Remark 12. We henceforth refer to the K– invariant norm described above as the Frobe-nius norm of the G–variety Hom(Γ, G) ⊂ ⊕rj=1MnC.Once it has been equipped with this additional structure, we can define the Kempf–Nessset M of Hom(Γ, G) as in Section 2.3.3 to be those representations that are of minimalnorm within their G– orbit. The Neeman–Schwarz Theorem now provides us with a K–equivariant deformation retraction of Hom(Γ, G) onto M.222.4.2 The Kempf–Ness setIn order to fruitfully apply the Neeman–Schwarz Theorem to Hom(Γ, G), we need to un-derstand the target of the retraction it provides. For a general group Γ, e.g., if Γ is a freegroup, explicitly determining the Kempf–Ness set of Hom(Γ, G) appears to be a hopelesstask. This is why we henceforth let Γ be a finitely generated nilpotent group. In this case,the Kempf–Ness set admits a very nice description:Theorem III. Let G ⊂ SLnC be a complex reductive algebraic group and suppose thatK = G ∩ SUn ⊂ G is a maximal compact subgroup. If Γ is a finitely generated nilpotentgroup and M denotes the Kempf–Ness set of the G–subvariety Hom(Γ, G) ⊂ ⊕rj=1MnCequipped with the Frobenius norm, thenM = {ρ ∈ Hom(Γ, G) : ρ(Γ) consists of normal matrices}.The proof of Theorem III relies on two lemmas. Before diving into the heart of thematter, we refresh the reader’s memory with some elementary linear algebra. Recall thata matrix m ∈ MnC is normal if mm∗ = m∗m or, equivalently, V has an orthogonalbasis of eigenvectors of m. The following well known1 characterization of normal matricesfollows from the unitary invariance of the Frobenius norm || · || of MnC and the SchurTriangularization Theorem.Lemma 2.4.1. If λ1, . . . , λn denote the eigenvalues of the matrix m ∈MnC, then||m||2 ≥n∑j=1|λj |2.Moreover, equality holds if and only if m is a normal matrix.Our second lemma concerns the images of certain representations of Γ in G:Lemma 2.4.2. Let G ⊂ SLnC be a reductive algebraic group and suppose that K = G∩SUnis a maximal compact subgroup of G. If N ⊂ G is a nilpotent reductive algebraic subgroup,then there is some g ∈ G for which gNg−1 consists of normal matrices.Proof. Recall from Lemma 2.2.4 that a reductive nilpotent algebraic group N has a uniquemaximal compact subgroup C. Since C is contained in a maximal compact subgroup of G1We invite the intrigued reader to consult Grone–Johnson–Sa–Wolkowicz [26] where this is the 53rd (outof 70!) characterization of normality.23and all such groups are conjugate in G, there is some g ∈ G for which gCg−1 ⊂ K. Wecan therefore assume (after conjugating by g) that N ⊂ G ⊂ SLnC and C ⊂ K ⊂ SUn.Being a reductive algebraic group, N is the complexification of C. Thus, it admits apolar decomposition (c.f. Section 2.2.4):N = C · exp(ic) ' C × exp(ic)where exp(ic) ⊂ N◦ is central in N by Lemma 2.2.4. This shows that any matrix m ∈ Ncan be written uniquely as a product m = k · h for some unitary k ∈ C and Hermitianh ∈ exp(ic). Since unitary and Hermitian matrices are normal and since the product oftwo commuting normal matrices is normal, N consists of normal matrices.The first lemma is used to show that normal representations are critical points ofthe Kempf–Ness functions so they belong to the Kempf–Ness set. Then, the second lemmashows that polystable representations become normal after conjugation so the Kempf–Nessset consists exclusively of normal representations. More precisely:Proof of Theorem III. Let ρ ∈ Hom(Γ, G) and consider the corresponding tuple(m1, . . . ,mr) := (ρ(γ1), . . . , ρ(γr)) ⊂ Gr.If mj is a normal matrix for each 1 ≤ j ≤ r, then Lemma 2.4.1 ensures that||(m1, . . . ,mr)|| ≤ ||(gm1g−1, . . . , gmrg−1)||for any g ∈ G. Consequently, the representation ρ is in M; this shows thatM⊃ {ρ : ρ(γj) is normal for 1 ≤ j ≤ r} ⊃ {ρ : ρ(Γ) consists of normal matrices}.We now show that the three sets coincide. Seeking a contradiction, suppose that ρ ∈M ⊂Hom(Γ, G) ps is a representation whose image doesn’t consist of normal matrices. Recallthat Richardson’s Theorem characterizes polystable points in Gr for the conjugation actionof G as those tuples (m1, . . . ,mr) for which A (m1, . . . ,mr) ⊂ G is reductive. Therefore,the Zariski closure of the image of our representationN := A (ρ(Γ)) = A (ρ(γ1), . . . , ρ(γr))24is a reductive nilpotent algebraic subgroup of G. But then, Lemma 2.4.2 produces anelement g ∈ G for which gNg−1 consists of normal matrices and, by Lemma 2.4.1, thiscontradicts the assumption that ρ was of minimal norm within its G– orbit.Example 6. While it is more or less clear that if we replace Γ by a free group thenM doesnot consist of normal representations (see, for instance, Florentino–Lawton [24, Remark3.2]), it may be somewhat surprising that this already fails to be the case when Γ is solvable.To see this, consider the solvable group Γ := Z o Z/4Z generated by the elements (1, 0)and (0, 1). For this generating set, an obvious representation ρ ∈ Hom(Γ, SL2C) is givenby mapping(1, 0) 7→(λ 00 λ−1)and (0, 1) 7→(0 1−1 0)where |λ| 6= 1, 0. Since both of these matrices are normal, it follows that ρ lies in theKempf-Ness set. Nevertheless, their product is not a normal matrix! This example wasshown to us by Lior Silberman.2.5 The scaling operationIn the previous section, we determined the Kempf–Ness set of the representation varietyHom(Γ, G). To complete the proof of Theorem I in the complex case, we seek a deformationretraction of this set onto Hom(Γ,K ). The goal of this section is to show that scalingeigenvalues does the trick.2.5.1 Scaling eigenvaluesFollowing Pettet and Souto [43], we define an eigenvalue scaling operator on the semisimpleelements of an algebraic group. Let Dn denote the diagonal subgroup of SLnC, identifiedin the usual way with a subgroup of (C×)n. Consider the mapσ˜ : [0, 1]× Dn → Dn, (t, (λi)) 7→ σ˜t((λi)) := (e−t log |λi|λi)where log(·) denotes the standard real logarithm. Notice that σ˜ is continuous, equivariantunder the action of the Weyl group, and that σ˜0(g) = g while σ˜1(g) ∈ Dn ∩ SUn for allg ∈ Dn.Given now a semisimple element g ∈ SLnC, there is some h ∈ SLnC for which hgh−1 ∈25Dn. Since σ˜ is invariant under the Weyl group, we see that for all t ∈ [0, 1] the elementσt(g) := h−1σ˜t(hgh−1)his independent of the choice of h. In particular, we obtain a well defined scaling mapσ : [0, 1]× (SLnC)s → (SLnC)s, (t, g) 7→ σt(g) (2.4)where (SLnC)s denotes the set of semisimple elements in SLnC. This map preserves com-mutativity and scales the eigenvalues of the semisimple elements of SLnC until they landin a subgroup conjugate to SUn. Using this map, the following lemma is proved in [43,Section 8.3]:Lemma 2.5.1 (Pettet–Souto). If G is any linear algebraic group and K ⊂ G is a maximalcompact subgroup, then there is a G–equivariant continuous mapσ : [0, 1]×Gs → Gs, (t, g) 7→ σt(g)which satisfies the following properties for all g, g1, g2 ∈ Gs:1. σ0(g) = g and σ1(g) is conjugate to an element of K.2. ∀ 0 ≤ t ≤ 1, σt fixes pointwise elements of G which are conjugate to elements of K.3. ∀ 0 ≤ t ≤ 1, if g1g2 = g2g1 then σt(g1g2) = σt(g1)σt(g2) = σt(g2)σt(g1).4. If G is defined over R then σ is equivariant under complex conjugation.2.5.2 Scaling the Kempf–Ness setLet us now return to the setting of Theorem I where Γ is a finitely generated nilpotent groupand G is a complex reductive algebraic group. In Section 2.4.1, given a maximal compactsubgroup K ⊂ G, we chose a faithful representation G ↪→ SLnC for which K = G∩SUn toendow Hom(Γ, G) ⊂ ⊕rj=1MnC with a K– invariant Frobenius norm. This allowed us todefine the Kempf–Ness setM of Hom(Γ, G) as the set of representations of minimal normwithin their G– orbit. In fact, Theorem III showed thatM = {ρ ∈ Hom(Γ, G) : ρ(Γ) consists of normal matrices}.26The goal of this subsection is to use the scaling map σ from Lemma 2.5.1 to prove thefollowing proposition:Proposition 2.5.2. Keeping the notation as above, there is a K– equivariant strong de-formation retraction of the Kempf–Ness set M onto Hom(Γ,K ).Before proving the proposition, we need to understand how the image of representationsin the Kempf–Ness set behave under scaling; this is the content of the following lemma.Lemma 2.5.3. Let N ⊂ SLnC be a nilpotent reductive algebraic group. If σt denotes thescaling map from Lemma 2.5.1, then the restriction of σt to N is a group homomorphism.Proof. Recall from Lemma 2.2.4 that a nilpotent reductive algebraic group N consists ofsemisimple elements and has a unique maximal compact subgroup C. Moreover, it admitsa polar decompositionN = C · exp(ic) ' C × exp(ic)where c denotes the Lie algebra of C and exp(ic) ⊂ N◦ is central in N by Lemma 2.2.4again.Let m1 = k1 · h1 and m2 = k2 · h2 denote the unique decomposition of elementsm1,m2 ∈ N into products of elements k1, k2 ∈ C and h1, h2 ∈ exp(ic). Notice that sinceC is maximal compact it is fixed pointwise by σt. Then, since h1 and h2 are central in N ,we have:σt(m1m2) = σt(k1h1k2h2) = σt(k1k2h1h2) = σt(k1k2)σt(h1)σt(h2) = k1k2σt(h1)σt(h2)= k1σt(h1)k2σt(h2) = σt(k1)σt(h1)σt(k2)σt(h2) = σt(k1h1)σ(k2h2) = σt(m1)σt(m2).Here, we repeatedly use parts (2) and (3) of Lemma 2.5.1.Proof of Proposition 2.5.2. Let us denote by N the set of normal matrices in SLnC andconsider the restriction of σt to N . Observe that σ0(N ) = N while σ1(N ) = SUn becausea matrix is unitary if and only if it is normal and its eigenvalues all have unit norm. Sincethe representations ρ in the Kempf–Ness setM of Hom(Γ, G) are precisely those for whichN := A (ρ(Γ)) ⊂ N is a reductive nilpotent group, it follows from Lemma 2.5.3 that σt ◦ ρis also inM. Consequently, the map ψ :M× [0, 1]→M where ψ(ρ, t) := σt ◦ρ is a strongdeformation retraction with ψ(ρ, 1)(Γ) ⊂ K = G ∩ SUn.272.6 The real and complex casesThe previous two sections contain a complete proof of Theorem I when G is a complexreductive linear algebraic group. To recapitulate, once Hom(Γ, G) has been endowed withthe appropriate structure, the retraction proceeds essentially in two steps:Hom(Γ, G)Neeman−Schwarz−−−−−−−−−−−−→TheoremKempf–Ness SetScaling−−−−−−−−→EigenvaluesHom(Γ,K ).In this section, we use real Kempf–Ness theory to adapt these steps when G is replaced byits group of real points G(R).2.6.1 Real Kempf–Ness theoryTo begin, we discuss the work of Richardson and Slodowy found in [47] where they developreal analogues of the main results in Kempf–Ness theory.Let V be a finite dimensional complex vector space with real structure V (R) and letH be a positive-definite Hermitian form on V . In this case, we always have H(x, y) =S(x, y) + iA(x, y) where S (respectively A) is a symmetric (respectively alternating) real-valued R-bilinear form on V . We say that the Hermitian form H is compatible with theR−structure of V if A vanishes on V (R)× V (R).Example 7. The Frobenius inner product on MnC is compatible with the R−structureMnR.Let us suppose now that the complex reductive algebraic group G ⊂ GL(V ) is definedover R and denote its group of real points by G(R). Since G is reductive, it follows fromthe work of Mostow [38] that, upon identifying V ∼= Cm and GL(V ) ∼= GLmC, we canassume G to be stable under the involution θ(g) := (g∗)−1. In this case, the fixed pointsK := Gθ = {g ∈ G : θ(g) = g}form a maximal compact subgroup of G and the real fixed points K(R) = G(R)θ forma maximal compact subgroup of G(R). One can show [47, Appendix 2] that the vectorspace V may always be equipped with a K– invariant Hermitian inner product which iscompatible with the R−structure V (R). We denote the associated K– invariant norm onV by || · ||.28The (complex) Kempf–Ness set M of V for the action of G can now be defined as inSection 2.3.2 and the (real) Kempf–Ness set M(R) of V (R) for the action of G(R) can bedefined by the analogous ruleM(R) := {v ∈ V (R) : ||g · v|| ≥ ||v|| for all g ∈ G(R)}. (2.5)The following lemma is an immediate consequence of [47, Lemma 8.1]:Lemma 2.6.1. M(R) =M∩ V (R).At this point we can state the two only results of Richardson and Slodowy that we shallneed. The first is an analogue of Corollary 2.3.2 [47, Theorem 7.7]:Theorem 2.6.2. Keeping the notation as above, let X be a closed G(R)−stable subsetof V (R). If MX(R) := M(R) ∩ X, then there is a homeomorphism MX(R)/K(R) ∼=X//topG(R).Remark 13. Here, X//topG(R) denotes the topological Hilbert quotient (c.f. Section 2.3.1and [47, Section 7.2]).The second is an analogue of the Neeman–Schwarz Theorem [47, Theorem 9.1]:Theorem 2.6.3. Keeping the notation as above, there is a continuous K(R)−equivariantstrong deformation retraction ϕ : V (R) × [0, 1] → V (R) of V (R) onto M(R). Moreover,the deformation is along orbits of G(R), that is, ϕt(v) ⊂ G(R) · v for 0 ≤ t < 1 andϕ1(v) ∈ G(R) · v.Remark 14. As in the complex case, if X is a closed G(R)-stable subset of V (R), then therestriction of ϕ to X × [0, 1] is a deformation retraction of X onto MX(R) :=M(R) ∩X.2.6.2 Proofs of Theorem I and Theorem IIWe are now ready to complete the proofs of our main theorems. We begin with the proofof the real case of Theorem I:Theorem I. Let Γ be a finitely generated nilpotent group and let G be the group of complexor real points of a reductive linear algebraic group defined over R. If K is a maximalcompact subgroup of G, then there is a K– equivariant strong deformation retraction ofHom(Γ, G) onto Hom(Γ,K ).29Proof of the real case. Let G ⊂ SLnC be a complex reductive algebraic group defined overR and denote its group of R−points by G(R). In view of the Conjugacy Theorem 2.2.1, itis no loss of generality to assume that K = G ∩ SUn is a maximal compact subgroup of Gfor which K(R) = K ∩G(R) is the maximal compact subgroup of G(R) in the statementof Theorem I. We can then proceed essentially as in Section 2.4.1. If Γ is generated byr elements, we embed Hom(Γ, G(R)) ⊂ ⊕rj=1MnR and Theorem 2.6.3 provides us with aK(R)−equivariant strong deformation retraction of Hom(Γ, G(R)) onto its Kempf–Nessset M(R). Here, Lemma 2.6.1 and Theorem III imply thatM(R) = {ρ ∈ Hom(Γ, G(R)) : ρ(Γ) consists of normal matrices }.The proof is then completed by Lemma 2.5.1 and Proposition 2.5.2 which show that scalingeigenvalues induces a K(R)−equivariant retraction of M(R) onto Hom(Γ,K(R)).Finally, we prove the analogous theorem for character varieties:Theorem II. Let Γ be a finitely generated nilpotent group and let G be the group of complexor real points of a reductive linear algebraic group, defined over R in the latter case. IfK ⊂ G is any maximal compact subgroup, then there is a strong deformation retraction ofthe character variety Hom(Γ, G)//G onto Hom(Γ,K )/K.Proof. LetG ⊂ SLnC be a complex reductive linear algebraic group and equip Hom(Γ, G) ⊂Gr with the Frobenius norm as in Section 2.4.1. If we letM denote its Kempf–Ness set, thenCorollary 2.3.2 shows that there is a homeomorphism Hom(Γ, G)//G ∼=M/K. Proposition2.5.2 then produces a K–equivariant strong deformation retraction ofM onto Hom(Γ,K )which induces a strong deformation retraction of M/K onto Hom(Γ,K )/K. If G is de-fined over R, the argument above can be carried out for its group of real points G(R) byviewing Hom(Γ, G(R))//topG(R) as a topological Hilbert quotient (c.f. Section 2.3.1 andRichardson–Slodowy [47, Section 7.2]) and invoking Theorem 2.6.2 instead of Corollary2.3.2.2.7 Expanding nilpotent groupsIn this last section of the chapter, we complete the comment made in the introduction onthe necessity of the assumption that G be reductive in Theorem I. The following content30was shown to us by Lior Silberman and Juan Souto. We refer the reader to Raghunathan[44, Chapter 2] for generalities on nilpotent lattices in unipotent Lie groups.Recall from the work of Mal’cev that every finitely generated torsion free nilpotentgroup admits a canonical completion to a unipotent Lie group which is usually called itsMal’cev completion. More precisely, one has the following theorem:Theorem 2.7.1 (Mal’cev). Let Γ be a finitely generated torsion free nilpotent group. Then,there is a canonical unipotent Lie group U for which Γ ⊂ U is a lattice. Moreover, ifU ′ is any unipotent Lie group, then any homomorphism Γ → U ′ extends uniquely to ahomomorphism U → U ′.Our goal is to show that one can choose Γ and U in a way that makes the conclusionof Theorem I fail when we consider the space of representations Hom(Γ, U ). It turns outthat the key property of Γ ⊂ U in this setting is the “richness” of its automorphism groupas described in the following definition.Definition 2.7.2. A finitely generated torsion free nilpotent group Γ is said to be expandingif the Lie algebra u of its Mal’cev completion U admits a semisimple automorphism, all ofwhose eigenvalues have norm larger than 1.Example 8. Free nilpotent groups and Heisenberg groups are expanding.The first non-expanding torsion free nilpotent group was implicitly produced by Dixmierand Lister in [20]. Later, Dyer [22] constructed a unipotent group U with a unipotentautomorphism group Aut(U ). Our interest in Dyer’s counterexample stems from the factthat, being unipotent, Aut(U ) ⊂ Hom(U,U ) is a closed subset. Since it is also an opensubset, we can conclude that Hom(U,U ) is a disconnected topological space. Moreover,since the Lie algebra of U has rational structure constants, there is a (nilpotent) latticeΓ ⊂ U and the results of Mal’cev mentioned above ensure that Hom(Γ, U ) = Hom(U,U ).However, since the maximal compact subgroup K of a unipotent group U is always trivial,Hom(Γ,K ) consists of a single point. In particular, Hom(Γ,K ) is not homotopy equivalentto Hom(Γ, U ).On the other hand, when Γ is an expanding nilpotent group the situation is entirelydifferent. If we let U be the Mal’cev completion of Γ and U ′ be any unipotent Lie group,the representation space Hom(Γ, U ′) = Hom(U,U ′) is contractible. Indeed, since Aut(U )is an algebraic group, it has finitely many connected components. If σ is an expanding31automorphism of Γ, it follows that σl ∈ Aut(U )◦ for some positive integer l. We cannow choose a one-parameter-subgroup of automorphisms of Γ containing σl and reversingthe associated flow induces the desired contraction. This is the observation that allowedSilberman and Souto to prove Theorem I for expanding nilpotent groups.32Chapter 3A Note on NilpotentRepresentationsThis chapter is a transcript of [8] consisting of joint work by the author with Lior Silberman.3.1 IntroductionLet G be the group of complex points of an affine algebraic group. When Γ is a finitelygenerated group, one may parametrize the homomorphisms from Γ to G by the images ofa finite generating set. This realizes Hom(Γ, G) as an (affine) algebraic set, carved out of afinite product of copies of G by the relations of Γ. As a complex variety, Hom(Γ, G) admitsa natural Hausdorff topology obtained from an embedding into affine space and it is easyto see (and well-known) that the analytic space structure on Hom(Γ, G) is independent ofthe chosen presentation of Γ. Here, we will only consider the case where G is reductivethough, in principle, the questions we address below can be asked without this assumption.These spaces of homomorphisms are of classical interest (see Lubotzky–Magid [35] andthe references therein) and their algebraic topology has been the subject of much recentscrutiny (see, for instance, [3–6, 18, 19, 25]), stemming in part from the work of A´dem andCohen [2]. In this context, it was recently shown by the first named author [7] that if Γ isnilpotent and K is a maximal compact subgroup of G, then there is a strong deformationretraction of Hom(Γ, G) onto Hom(Γ,K ). This result was first established by homotopy-theoretic methods for Γ abelian by Pettet and Souto [43] and for Γ expanding nilpotent bySouto and the second named author. The result for arbitrary nilpotent groups was obtained33in [7] by replacing these earlier approaches with algebro-geometric methods. Nevertheless,the machinery developed by Pettet–Souto and its followups is very well posed to the studyof topological invariants. Accordingly, the goal of this note is to combine these topologicaland algebro-geometric tools to obtain topological information about representation spacesof nilpotent groups.From now on, fix a non-abelian finitely generated s-step nilpotent group Γ. Recall thatthis means that the lower central series, defined inductively byΓ(1) = Γ, Γ(i+1) = [Γ,Γ(i)]has Γ(s) non-trivial but Γ(s+1) = {e}. The epimorphism Γ→ Γ/Γ(i) induces an embeddingHom(Γ/Γ(i), G)→ Hom(Γ, G)which (for general groups Γ and G) is not even an open map. Nevertheless, we will show:Theorem IV. Let Γ be a finitely generated nilpotent group and let G be the group ofcomplex points of a (possibly disconnected) reductive algebraic group. For all i ≥ 2, theinclusionHom(Γ/Γ(i), G)ι−→ Hom(Γ, G)is a homotopy equivalence onto the union of those components of the target intersecting theimage of ι.Consider Hom(Γ, G) as a based space by taking the trivial representation as the basepoint. In this case, Theorem IV implies that the connected componentsHom(Γ, G)1 ⊂ Hom(Γ, G) and Hom(Γ/Γ(i), G)1 ⊂ Hom(Γ/Γ(i), G)of the trivial representation are homotopy equivalent for all i ≥ 2. Using this, we willdescribe the homotopy type of the component of the trivial representation in terms ofabelian representations:Corollary 3.1.1. For Γ and G as in Theorem IV, there is a homotopy equivalenceHom(Γ, G)1 ' Hom(Z rankH1(Γ;Z), G)1.To introduce the other space we study, note first that the action of G on itself by34conjugation induces an action on Hom(Γ, G) and conjugate homomorphisms are oftenconsidered equivalent (this is the usual notion of equivalence of representations in GLnC).Accordingly one often wishes to understand the associated quotient but, unfortunately, thenaive topological quotient is not a nice space: it need not even be Hausdorff. In order to“repair” this space, we use the affine geometric invariant theory quotient Hom(Γ, G)//Ginstead. This so-called character variety is usually endowed with the structure of anaffine variety but, for our purposes, it may be constructed topologically as the universalquotient in the category of Hausdorff spaces (see Brion–Schwarz [16]). The systematicstudy of the topology of theses spaces has seen much recent development (see, for instance,[10, 11, 23, 24, 33]). Concentrating on the component of the trivial representation, we willuse Corollary 3.1.1 to prove:Corollary 3.1.2. Let Γ be a finitely generated nilpotent group and let G be the group ofcomplex points of a reductive algebraic group. Then1. pi1(Hom(Γ, G)1) ∼= pi1(G)rankH1(Γ;Z),2. pi1((Hom(Γ, G)//G)1) ∼= pi1(G/[G,G])rankH1(Γ;Z).Corollary 3.1.3. Let G be the group of complex points of a connected reductive algebraicgroup, let T ⊂ G be a maximal algebraic torus and let W be the Weyl group of G. If Γ isa finitely generated nilpotent group and F is a field of characteristic 0 or relatively primeto the order of W , then:1. H∗(Hom(Γ, G)1;F ) ∼= H∗(G/T × T rankH1(Γ;Z);F )W ,2. H∗((Hom(Γ, G)//G)1;F ) ∼= H∗(T rankH1(Γ;Z);F )W .While the results above indicate many similarities between representation spaces ofabelian and non-abelian nilpotent groups, the latter have a much richer topology thanthe former. For instance, recall that for a connected semisimple group S, the varietyHom(Z2, S) is irreducible and thus connected [45]. Moreover, Hom(Zr, SLnC), Hom(Zr, SpnC)and the corresponding character varieties are connected for all values of r and n. The sit-uation for non-abelian nilpotent groups is markedly different:Theorem V. Let G be the group of complex points of a (possibly disconnected) reductivealgebraic group. If Γ is a finitely generated nilpotent group which surjects onto a finite non-abelian subgroup of G, then Hom(Γ, G) and Hom(Γ, G)G are both disconnected topologicalspaces.35Since non-abelian free nilpotent groups and Heisenberg groups surject onto the non-abelian nilpotent group of order 8, this implies:Corollary 3.1.4. Let Γ be a non-abelian free nilpotent group or a Heisenberg group. If G isthe group of complex points of a reductive algebraic group, then Hom(Γ, G) and Hom(Γ, G)G are connected if and only if G is an algebraic torus.Remark 15. All of the preceding statements remain true when G is replaced by a compactLie groupK. In fact, we will prove most of them in this setting before obtaining the complexreductive case via a homotopy equivalence.Outline of the chapter. We begin Section 3.2 by describing compact representationspaces using a fibre bundle. Then, in Section 3.3, we use this bundle to prove Theorem IVand Theorem V along with their various corollaries.3.2 An interesting bundleThe goal of this section is to prove the following key proposition:Proposition 3.2.1. Let K be a (possibly disconnected) compact Lie group. If Γ is ans-step nilpotent group with s ≥ 2, then the set of abelian groupsF := {ρ(Γ(s)) ⊂ K : ρ ∈ Hom(Γ,K)}admits a homogeneous manifold structure with finitely many connected components forwhich the projection mapp : Hom(Γ,K)→ F , p(ρ) = ρ(Γ(s)) (3.1)is a locally trivial fibre bundle.The proof of Proposition 3.2.1 relies on the following lemma:Lemma 3.2.2. For all m ∈ N there is an O = O(m) ∈ N such that, if N ⊂ SUm is ans-step nilpotent group with s ≥ 2, then N(s) is an abelian subgroup of SUm of order boundedby O.Proof. Recall that N(s) is an abelian subgroup of SUm contained in the centre of N . Assuch, there is a direct sum decomposition Cm = V1⊕ . . .⊕ Vr and r characters χ1, . . . , χr :36N(s) → C× such that χi 6= χj for all i 6= j and γ(v) = χi(γ) · v for all γ ∈ N(s) and v ∈ Vi.Moreover, for all g ∈ N , γ ∈ N(s) and v ∈ Vi, we haveγ(g(v)) = g(γ(v)) = g(χi(γ) · v) = χi(γ) · g(v).This allows us to consider the restrictions of the determinant homomorphismdeti : N → C×, deti(g) := det(g|Vi)where, since C× is abelian and s ≥ 2, the subgroup N(s) must be contained in ker(deti).This means that for all i and all γ ∈ N(s), we havedeti(γ) = χi(γ)dimVi = 1,so χi(γ) is always a root of unity of order bounded by m. Consequently, N(s) is conjugatein SUm to a subset of those diagonal matrices whose diagonal elements are roots of unityof order bounded by m. This completes the proof since the order of this finite set does notdepend on s.Proof of Proposition 3.2.1. Choose a faithful embedding of K into SUm. By Lemma 3.2.2,there is a constant O ∈ N uniformly bounding the order of abelian subgroups of K occurringas the image of Γ(s) under homomorphisms ρ : Γ→ K. In order to give F a homogeneousmanifold structure, we first consider the slightly larger setF˜ := {A ⊂ K : A is an abelian subgroup of order bounded by O}.Observe that K◦ (the identity component of K) acts by conjugation on F˜ with closed sta-bilizers. As such, we can endow F˜ with the orbifold structure with respect to which eachK◦-orbit is a connected homogeneous K◦-manifold (see Onishchik-Vinberg [42]). Con-cretely, if we define the “connected normalizer” as NK◦(H) := NK(H) ∩ K◦, then theconnected component of H ∈ F˜ is identified with K◦/NK◦(H). Having a topology on eachK◦-orbit, we endow F˜ with the disjoint union topology. Since K is a compact Lie group,there are only finitely many conjugacy classes of abelian subgroups of K of order boundedby O (c.f. Mostow [39, Lemma 1]) and, in particular, F˜ has only finitely many connectedcomponents.37A homomorphism ρ : Γ(s) → K need not extend to the full group Γ so the mapp : Hom(Γ,K)→ F˜ , p(ρ) = ρ(Γ(s))may not be surjective. Accordingly, we denote F := p(Hom(Γ,K)) and observe by K◦-equivariance of p that it is a union of connected components of F˜ . Let Z ⊂ F denotethe connected component of a finite abelian subgroup H ∈ F and let H := p−1(Z) ⊂Hom(Γ,K). We can then identify the restriction of p to H with the twisted product(K◦ ×H)/NK◦(H)→ K◦/NK◦(H)where NK◦(H) acts on K◦ (resp. H) by right multiplication (resp. conjugation). Observingthat p−1(H) is closed in Hom(Γ,K), it follows that K◦ ·p−1(H) = H is closed and thereforealso open (since F has only finitely many components). This shows that p is a locally trivialfibre bundle.3.3 Proofs of the main resultsLet G be the group of complex points of a (possibly disconnected) reductive algebraicgroup and recall that such a G necessarily arises as the complexification of a (possiblydisconnected) compact Lie group K. In this section, we use Proposition 3.2.1 to prove theresults mentioned in the introduction. In most cases, we prove a corresponding statementwith K in lieu of G before obtaining the claimed result. We refer the reader to Onishchick–Vinberg [42] for basic facts about Lie groups and complex algebraic groups.Let Γ be an s-step nilpotent group with s ≥ 2 and recall that, for all i, the epimorphismΓ → Γ/Γ(i) induces an embedding Hom(Γ/Γ(i),K) → Hom(Γ,K). Often, we shall abusenotation and identify Hom(Γ/Γ(i),K) with its image under this embedding. As a firstconsequence of Proposition 3.2.1 we obtain:Proposition 3.3.1. Let K be a (possibly disconnected) compact Lie group. If Γ is a finitelygenerated nilpotent group then, for all i ≥ 2, the inclusionHom(Γ/Γ(i),K)ι−→ Hom(Γ,K)is a homeomorphism onto the union of those components of the target intersecting theimage of ι.38Proof. We proceed by induction on the nilpotence step of Γ. Recall from Proposition 3.2.1thatp : Hom(Γ,K)→ F , p(ρ) = ρ(Γ(s))is a locally trivial bundle. If Γ is 2-step nilpotent, then the image of ι consists of allrepresentations factoring through the abelianization of Γ, that is those such that ρ(Γ(2)) ={eK}. Since eK is fixed by the conjugation action of K, the subgroup {eK} ∈ F isan isolated point in the given topology. Thus, for any ρ ∈ p−1(eK), the full connectedcomponent of ρ (which is path-connected) has trivial restriction to Γ(2) and we see thatp−1 ({eK}) is the union of the connected components it intersects, completing the proof inthis case.Suppose now that Γ is s-step nilpotent. If i = s, the same argument as for the basecase applies. Otherwise, i < s and thenΓ/Γ(i) ∼= (Γ/Γ(s))/(Γ(i)/Γ(s))where the nilpotence step of (Γ/Γ(s)) is s − 1. As such, the induction hypothesis impliesthat each of the following two embeddingsHom(Γ/Γ(i),K)→ Hom(Γ/Γ(s),K)→ Hom(Γ,K)is a homeomorphisms onto those components of the target intersecting its image and,consequently, that the same holds for their composition.Proof of Theorem IV. The theorem follows at once by [7, Theorem I].We can now prove:Corollary 3.3.2. If Γ and K are as in Proposition 3.3.1, then there is a homeomorphismHom(Γ,K)1 ∼= Hom(Z rankH1(Γ;Z),K)1.Proof. By Proposition 3.3.1, we have a homeomorphismHom(Γ,K)1 ∼= Hom(H1(Γ;Z),K)1.Since H1(Γ;Z) = Γ/[Γ,Γ] is a finitely generated abelian group, we may identify H1(Γ;Z)39with Zr ⊕ A where r := rankH1(Γ;Z) and A is a finite abelian group. At this pointwe would like to show that Hom(Zr ⊕ A,K)1 = Hom(Zr,K)1. Seeking a contradiction,suppose that ρ0 ∈ Hom(Zr ⊕ A,K)1 maps A non-trivially into K. By assumption, thereis a continuous path of representations [0, 1] 7→ ρt starting at ρ0 and ending at the trivialrepresentation ρ1 = 1. But now, this path induces a continuous deformation in Hom(A,K)of the representation ρ0|A to the trivial representation. This is impossible since Lie groupscontain no small subgroups (c.f. Bredon [14, Corollary 0.4.5]).Proof of Corollary I. The corollary follows at once by Theorem I.Using this, we immediately obtain:Corollary II. Let G be the group of complex points of a reductive algebraic group. If Γ isa finitely generated nilpotent group, then:1. pi1(Hom(Γ, G)1) ∼= pi1(G) rankH1(Γ;Z), and2. pi1((Hom(Γ, G)//G)1) ∼= pi1(G/[G,G]) rankH1(Γ;Z).Proof. The two formulas follow at once from Corollary I by the main results of Go´mez–Pettet–Souto [25] and Biswas–Lawton–Ramras [11].In order to prove our second corollary, we need the following:Lemma 3.3.3. If K is a compact Lie group and Γ is a finitely generated nilpotent group,then Hom(Γ,K)1/K = (Hom(Γ,K)/K)1. In particular, Hom(Γ,K) is connected if andonly if Hom(Γ,K)/K is connected.Proof. Recall from Corollary 3.3.2 that any ρ ∈ Hom(Γ,K)1 factors through the torsionfree part of H1(Γ;Z). As such, by Baird [6, Lemma 4.2], ρ ∈ Hom(Γ,K)1 if and onlyif there is a torus T ⊂ K such that ρ(Γ) ⊂ T . Since this property is preserved underconjugation by elements of K, it follows that (Hom(Γ,K)/K)1 coincides with the quotientHom(Γ,K)1/K.We can now prove the cohomological formulas mentioned in the introduction.Corollary III. Let G be the group of complex points of a connected reductive algebraicgroup, let T ⊂ G be a maximal algebraic torus and let W be the Weyl group of G. If Γ isa finitely generated nilpotent group and F is a field of characteristic 0 or relatively primeto the order of W , then:401. H∗(Hom(Γ, G)1;F ) ∼= H∗(G/T × T rankH1(Γ;Z);F )W , and2. H∗((Hom(Γ, G)//G)1;F ) ∼= H∗(T rankH1(Γ;Z);F )W .Proof of Corollary III. Following Pettet–Souto [43, Corollary 1.5], let K ⊂ G be a maximalcompact subgroup such that TK := T ∩K is a maximal torus in K. Notice that, for anyr ∈ N,K/TK × T r → G/T × T ris a W -equivariant homotopy equivalence and, in particular, thatH∗(K/TK × T r)W ∼= H∗(G/T × T r)W . (3.2)Here, it follows from Baird [6, Theorem 4.3] that the left hand side of the equation isisomorphic to H∗(Hom(Zr,K)1). Now, letting r := rankH1(Γ;Z), our first formula followsat once from the homotopy equivalencesHom(Γ, G)1 ' Hom(Zr, G)1 ' Hom(Zr,K)1provided by Corollary I and Theorem I. Finally, it is also due to Baird [6, Remark 4] thatHom(Zr,K)1/K ∼= T rK/W so our second formula follows from the homotopy equivalenceand homeomorphisms(Hom(Γ, G)//G)1 ' (Hom(Γ,K)/K)1 ∼= Hom(Γ,K)1/K ∼= Hom(Zr,K)1/K.provided by Theorem II, Lemma 3.3.3 and Corollary 3.3.2.Remark 16. The homotopy types of distinct components of representation spaces aretypically different. For instance, if we take Γ to be the discrete Heisenberg group H3(Z),then Hom(Γ,SL2C) decomposes into a simply-connected component and a non simply-connected component. In fact, this phenomenon already occurs for Γ abelian as illustratedin Go´mez–Adem [4] and Go´mez–Pettet–Souto [25].Theorem V. Let G be the group of complex points of a reductive algebraic group. If Γ isa finitely generated nilpotent group which surjects onto a finite non-abelian subgroup of G,then Hom(Γ, G) and Hom(Γ, G)//G are both disconnected.Proof. Let ψ : Γ → N be a surjective homomorphism onto a finite non-abelian subgroupof G and let K be a maximal compact subgroup of G containing N . Notice in particular41that ψ ∈ Hom(Γ,K) ⊂ Hom(Γ, G). Since Hom(Γ,K) ' Hom(Γ, G) and Hom(Γ,K)/K 'Hom(Γ, G)//G by Theorem I and Theorem II, and since Hom(Γ,K) is disconnected if andonly if Hom(Γ,K)/K is disconnected by Lemma 3.3.3, it suffices to prove that Hom(Γ,K)is disconnected.Seeking a contradiction, suppose that Hom(Γ,K) is connected and recall from Propo-sition 3.3.1 that, in this case, Hom(Γ/Γ(i),K) is connected for all i ≥ 2. Choose a minimals ∈ N with the property that ψ(Γ(s+1)) = eK and denote the s-step nilpotent groupΓ/Γ(s+1) by Γˆ. If we consider the fibre bundle (c.f. Proposition 3.2.1)p : Hom(Γˆ,K)→ F , p(ρ) = ρ(Γˆ(s)),then p(ψ) = ψ(Γˆ(s)) 6= eK . As such, by Proposition 3.3.1 and our assumptions,ψ /∈ Hom(Γˆ,K)1 ∼= Hom(Γ/Γ(s+1),K)1 ∼= Hom(Γ,K)1 ∼= Hom(Γ,K)and this contradiction completes the proof.Corollary IV. Let Γ be a non-abelian free nilpotent group or a Heisenberg group. IfG is the group of complex points of a reductive algebraic group, then Hom(Γ, G) andHom(Γ, G)//G are connected if and only if G is an algebraic torus.Proof. If G is disconnected or not simply-connected then [43, Corollary 1.3], Theorem I,Theorem II and Lemma 3.3.3 show that Hom(H1(Γ;Z), G) and Hom(H1(Γ;Z), G)//G aredisconnected. As such, it suffices to consider the case where G is simply-connected. Noticethat such a G contains a subgroup isomorphic to SL2C and, since SL2C contains a copyof the non-abelian group Q of order 8 generated by the matrices(i 00 −i)and(0 1−1 0),so does G. Since Q is a Z/2Z central extension of Z/2Z × Z/2Z, it follows that if Γ iseither a non-abelian free nilpotent group or a Heisenberg group, then Γ surjects onto Q.The claim now follows from Theorem V.42Chapter 4ConclusionIn some sense, this thesis came to be through the author’s obsession with the work of Pettetand Souto [43] where it was first established that, for G a complex or real reductive linearalgebraic group and K ⊂ G a maximal compact subgroup, there is a strong deformationretraction of Hom(Zr, G) onto Hom(Zr,K). Prior to their result, most known topolog-ical invariants in this realm concerned spaces of representations into compact groups; itwas their deformation retraction which allowed many of these results to be extended torepresentations into reductive groups. This implicitly asked the following:Question 1. Given a real or complex reductive linear algebraic group G and a maximalcompact subgroup K ⊂ G, what are the classes of finitely generated groups Γ for whichthere is a deformation retraction of Hom(Γ, G) onto Hom(Γ,K)?Starting with abelian groups, there were then many natural directions in which onemight try to generalize Pettet and Souto’s result. Having established in the guise of The-orem I that nilpotent groups also offer a positive answer to Question 1, solvable groupsbecame the next natural class to investigate. In this context, although the key technicalstep in the proof of Theorem I breaks down (c.f. Example 6), we suspect that such a retrac-tion still exists. Indeed, together with Lior Silberman, we have the following tantalizingpartial result:Proposition I (Bergeron-Silberman). Let Γ be a finitely generated virtually solvable group,let G be the group of complex points of a (possibly disconnected) reductive linear algebraicgroup, and let K ⊂ G be a maximal compact subgroup. There is a constant O = O(G)such that if M denotes the Kempf-Ness set of the G-subvariety Hom(Γ, G) ⊂ ⊕rj=1MnC43with respect to a K−invariant norm, then the image of any representation in M consistsof semisimple elements and contains a normal abelian subgroup of index bounded by O.Proof. Recall from the Kempf-Ness Theorem that M is always contained in the set ofpolystable points of Hom(Γ, G). Moreover, recall from Richardson’s Theorem that if ρ ∈Hom(Γ, G) corresponds to the tuple (ρ(γ1), . . . , ρ(γr)), it is polystable if and only if thealgebraic subgroup of G generated by the elements {ρ(γ1), . . . , ρ(γr)} is reductive. We maytherefore assume without loss of generality that the Zariski closure of the image of anyrepresentation in M is a reductive solvable subgroup S ⊂ G. It then follows from workof Mal’cev [54, Lemma 3.5] that such an S contains a normal abelian subgroup of indexbounded by a constant O depending only on G. Finally, since the connected abelian groupS◦ is an algebraic torus and all unipotent elements of S must be contained in this identitycomponent, we conclude that S consists of semisimple elements.Interestingly, recalling from the Neeman-Schwarz Theorem that there is a deformationof Hom(Γ, G) ontoM, this already implies that when Γ is virtually solvable the connectedcomponent of the trivial representation Hom(Γ, G)1 ⊂ Hom(Γ, G) has the homotopy typeof the space of representations of a free-abelian group. Given that the latter’s topologyis relatively well understood, this paves the way for computations of various homotopyinvariants. Nevertheless, in order to understand why Proposition I is a truly exciting de-velopment, it is worth recalling the broad lines of Pettet and Souto’s initial argument in theabelian case. Their retraction proceeds in two steps. They first construct a commutativity-preserving deformation of G onto its set of semisimple elements Gs in order to induce adeformation of Hom(Zr, G) onto the set Hom(Zr, Gs) of representations whose images con-sist of semisimple elements. Then, using homotopy-theoretic tools, they manage to deformHom(Zr, Gs) onto Hom(Γ,K). Although it remains unclear whether the first step of theirproof is adaptable to the virtually abelian case, it would seem that the homotopy-theoretictechniques they developed (further generalized in unpublished work by Silberman andSouto for expanding nilpotent group representations) can be modified to treat this virtu-ally abelian case. Once combined with Proposition I, this would add virtually solvablegroups to the class of positive answers for Question 1.This first exploration in the case of nilpotent and solvable groups is also intriguing inthat it generalizes Pettet and Souto’s result from the point of view of both Lie theory andgeometric group theory. Pushing further in the latter direction, one can consider Question1 in the context of right-angled Artin groups (c.f. [17]) or more general amalgamations of44abelian groups. Although the author does not believe that there is a deformation retractionfrom Hom(Γ, G) onto Hom(Γ,K) for this class of groups, there is still much to be said onthis topic which we are currently investigating with Juan Souto. Indeed, we have recentlyshown that when the underlying graph of a right-angled Artin groups Γ is a tree thenthere is a deformation of Hom(Γ, G) onto Hom(Γ,K). In fact, we prove a more generalstatement for tree-like amalgamations of abelian groups. To be precise, recall that a finitetree of free abelian groups T consists of a finite tree T , a free abelian group Tv for everyvertex v of T , and a free abelian group Te for every edge e of T , together with an injectivehomomorphisms Te ↪→ Tv whenever the vertex v is an endpoint of the edge e. We canthen consider the fundamental group AT of the tree of free abelian groups T in the senseof Bass and Serre [50]. For instance, if T consists of a single edge e joining a vertex v to avertex w then AT ∼= Tv ∗Te Tw is the amalgamated product of Tv and Tw along Te. Whatwe prove in this context is the following:Theorem VI (Bergeron-Souto [9]). Let AT be the fundamental group of a finite tree offree abelian groups and let G be the group of complex or real points of a (possibly discon-nected) reductive linear algebraic group, defined over R in the latter case. If K ⊂ G is anymaximal compact subgroup, then there is a strong deformation retraction of Hom(AT , G)onto Hom(AT ,K ).One amusing consequence of Theorem VI is that it implies without much work that asimilar statement holds for right-angled Artin groups defined by a disjoint union of treesand triangles. This class of groups may seem a bit artificial at first sight but it turns outthat a result of Droms [21] characterizes them as those right-angled Artin groups being thefundamental groups of 3-manifolds. It should also be pointed out that, as far as the authorcan tell, no such retraction should exist for a large class of right-angled Artin groups. Moreconcretely, let us conclude with the following question which the author suspects to havea positive solution:Question 2. For every m ∈ N with m ≥ 5 is there some n ∈ N such that, if AΓ is theright-angled Artin group determined by an m-cycle, then there is no strong deformationretraction of Hom(AΓ, SLnC) onto Hom(AΓ, SUn)?45Bibliography[1] A. Adem and M. C. Cheng. Representation spaces for central extensions and almostcommuting unitary matrices. to appear in the Journal of the London MathematicalSociety (arXiv preprint arXiv:1502.05092), 2015. 3[2] A. Adem and F. R. Cohen. 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