Geometric retracts of Siegel’s upper half space by Justin Martel B.Sc., The University of Ottawa, 2011 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2013 c Justin Martel 2013 Abstract The purpose of this thesis is to construct a codimension 1 cocompact Sp2g Z-equivariant strong deformation retract Wg of Siegel’s upper half space hg . This yields a partial contribution towards the problem of constructing a strong spine for the real linear symplectic group Sp2g R. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 What is a spine? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The wellrounded retract for SLn . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Siegel’s upper half space hg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 2.2 Symmetric space structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Cartan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Killing form and the Ug -invariant metric . . . . . . . . . . . . . . . 7 2.1.3 Nonpositive curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.4 K¨ ahler structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.5 Iwasawa decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 9 Symplectic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Some symplectic linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Intersecting lagrangian subspaces 3.2 Heights of ω-orthogonals . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Lagrangian systole function . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Systoles on hg 4.1 11 iii Table of Contents 4.2 L -wellrounded lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 The equivariant retract Wg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6 Cocompactness of Wg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 8 Bavard’s retract Yg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 9 Weak and strong spines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 10 MacPherson and McConnell’s explicit reduction theory . . . . . . . . . 33 . . . . . . . . . . . . . . . . . . . . . . . . . . 33 10.2 Embedding h2 into S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 10.3 Barycentric retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7 Codimension of Wg 10.1 Voronoi’s cell decomposition 11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 iv Acknowledgements I am grateful to the Topology group at UBC, whose participants were always generous with their time and energy. I have been particularly inspired by my interactions with Lior Silberman, Juan Souto, and Alexandra Pettet - all of whom have contributed towards my development as a mathematician. Thank you. v Dedication To my parents - Andr´e and Debra. vi Chapter 1 Introduction 1.1 What is a spine? We are compelled to the topic of this thesis by the following consideration on the cohomology of arithmetic groups. Given a semisimple algebraic group G (e.g. SLn , Sp2g , . . .), the connected component of its R-points G = G(R)o yields a semisimple Lie group. For a maximal compact subgroup K, the quotient X = K\G is a finite-dimensional contractible nonpositively curved space. The subgroup of Z-points Γ = G(Z) has a properly discontinuous right action on K\G given by (Kx, γ) → Kxγ for x ∈ G, γ ∈ Γ. The discrete arithmetic subgroup Γ contains nontrivial finite subgroups, and hence does not act freely on the contractible space X, i.e. there are points Kx whose Γ-stabilizers are nontrivial. The presence of torsion elements means Γ = G(Z) has infinite cohomological dimension. However it has been known since Minkowski that Γ contains many finite index torsion-free subgroups Γ , e.g. the congruence subgroups Γ(N ) := ker (SLn Z → SLn (Z/N Z)) for N ≥ 3. These finite index torsion-free subgroups act freely on X, and thus have finite cohomological dimension with the space X serving as explicit model for the classifying space EΓ . One therefore considers Γ to virtually have finite cohomological dimension. For a given finite index torsion free subgroup Γ of Γ, the maximal integer q for which the cohomology module H q (Γ , Z) does not vanish gives the cohomological dimension cd Γ of Γ . One can show (c.f. [5]) that this integer q is constant for every finite-index torsion free subgroup of Γ, and thus we define q to be the virtual cohomological dimension vcd Γ of Γ. The virtual cohomological dimension thus describes the minimal possible dimension of a classifying space for the finite index torsion-free subgroups of Γ = G(Z). A remarkable paper of Borel-Serre [4] computes the virtual cohomological dimension of the semisimple 1 1.2. Outline of thesis algebraic groups as follows: vcd G(Z) = dimR K\G − rankQ G. Thus the formula expresses the vcd in terms of the dimension of the symmetric space minus the Q-rank of the group, where we recall that rankQ G is the dimension of the maximal Q-split torus in G. For instance SLn , Sp2g have rankQ = n − 1, 2g. The computation of Borel-Serre motivates us to formulate the following problem. Problem of the Spine. For a semisimple Lie group G with maximal compact K construct a subset W of K\G for which • W is a Γ-equivariant strong deformation retract of K\G; • Γ acts cocompactly on W; and • W has dimension equal to the virtual cohomological dimension vcd(Γ) of Γ. The existence of a spine for SL2 has probably been a long recognized fact (c.f. [5]). The question of constructing a spine is probably first due to Soul´e in his 1973 thesis [20]. Afterwords Ash [2], motivated by group cohomology, expanded Soul´e’s retraction and clearly formulated the wellrounded retract (see 1.3). 1.2 Outline of thesis The present thesis concerns itself with the question of constructing a spine for the linear symplectic group Sp2g and can only offer a partial solution. Our main object of study are symplectic lattices and their associated Teichm¨ ueller space hg . The basic facts on the symmetric space hg are described in 2. As preliminary to our main results we require some elementary facts concerning the symplectic linear algebra which are presented in 3. The important lagrangian systole function is introduced in 4. Our main result is contained in chapters 5, 6, 7, wherein we construct a cocompact Sp2g Z-equivariant strong deformation retract Wg having codimension 1. We present Bavard’s own codimension retract Yg in 8 which is essentially different from our own, and MacPherson and McConnell’s weak spine for Sp4 in 10. Finally, the general existence question for spines (weak and strong) is described in 9. 2 1.3. The wellrounded retract for SLn While our result is well-short of constructing a spine for hg (our original motivation), we do believe our retract to have some interesting features. In particular the retract is based on the lagrangian systole function sysL . This is the first instance of a retract being defined by a higher-dimensional systole function, where all previous retracts have been defined against the 1-systole function. Of course the lagrangian systole function is special to our lattices being symplectic. In 4 we establish some basic systolic properties of sysL (e.g. it is bounded from above on hg ) which we believe to have some independant interest for the systolic geometry of symplectic tori. These preliminary results naturally suggest some interesting questions on the existence of systolic inequalities within the symplectic geometry which we shall investigate in the future. 1.3 The wellrounded retract for SLn Before proceeding to our general problem, it is worthwhile to describe the wellrounded retract, i.e. a spine for the special linear group. In dimension two the symmetric space S2 := SO2 R\SL2 R is well known to coincide with the hyperbolic upper half space H. We see a point SO2 R · A ∈ S2 as describing an isometry class of marked 2-dimensional unimodular lattices Λ = SO2 R · AZ2 in R2 . It is convenient to write AZ2 ⊂ R2 and work SO2 R-equivariantly. An element in the arithmetic group SL2 Z acts on AZ2 by changing the marking, i.e. (AZ2 , γ) → AγZ2 . One should observe that γ acts as volume-preserving group automorphism on Z2 (that is, γZ2 = Z2 ). The 1 ±1 0 −1 and . arithmetic group SL2 Z is seen to be generated by the elements 0 1 1 0 From this, one readily identifies a fundamental domain for the action of SL2 Z on S2 to be given by F := {x + iy : y > 0 |x| ≤ 1/2, x2 + y 2 ≥ 1}. This is the ‘key-hole’ description of the so-called modular domain. The upper half arc of the unit circle in F, namely the set {x + iy ∈ F : x2 + y 2 = 1}, is a compact codimension 1 subspace in F whose SL2 Z-translates generate a tree X2 for S2 . The tree X2 (which also corresponds to the Bass-Serre tree for SL2 Z) is a spine for S2 . The equivariant retraction of S2 to X2 admits several descriptions. The first description of an equivariant strong deformation retract will generalize to the so-called wellrounded retract of Soul´e and Ash ([20], [2]) yielding a spine for the higher 3 1.3. The wellrounded retract for SLn dimensional symmetric spaces Sn , (n ≥ 2). It begins with interpreting the subset X2 as consisting of those 2-dimensional unimodular lattices Λ whose shortest nonzero vectors span R2 . It is well to observe that the length of this shortest nonzero vector is SL2 Z-invariant. For any Λ ∈ S2 \ X2 there will exist, up to a sign, a unique nonzero shortest vector. Call it v. Consider the rank-1 Q-split torus TΛ,ρ defined by expanding by ρ ≥ 1 along Rv, and contracting by ρ−1 along the orthogonal complement Rv o . There is a minimal time τΛ for which expanding along Rv will yield a new shortest nonzero vector. At this minimal time τΛ we have deformed the lattice Λ into a ‘wellrounded lattice’ Λ . For a lattice Λ ∈ X2 , we do nothing and leave it stationary, i.e. τΛ = 0. An important technical point is to demonstrate that the function τ : S2 → R is continuous. Then r : [0, 1] × S2 → S2 defined by r(s, Λ) := TΛ,(1−s)+sτΛ Λ is a well-defined SL2 Z-equivariant continuous strong deformation retract of S2 onto X2 . An alternative description, which is generalized to arbitrary semisimple algebraic groups by Borel-Serre [4] begins with the following observation. The flow generated by the wellrounded retract on the fundamental domain F can be seen to arise from the geodesic flow on H. More precisely consider the geodesic flow on the fundamental domain F. The wellrounded retract coincides with flowing each point along geodesics away from the cusp of the modular domain. The 2-dimensional spine X2 extends, as described by Soul´e and Ash to an SLn Zequivariant strong deformation retract Xn , where Xn consists of those unimodular ndimensional lattices whose shortest nonzero vectors (i.e. 1-systoles) span Rn . Using Mahler’s compactness criterion (c.f. 6) it is immediate that indeed Xn is cocompact, and moreover has codimension equal to rankQ SLn . The deformation of Sn onto Xn is known as the wellrounded retract. The emphasis on distinguishing Xn from the subset Xn consisting of lattices whose 1-systoles span the lattice as a Z-module is essential. Pettet and Souto [16] established that in fact Xn is a minimal cocompact equivariant retract of Sn . In particular the subset Xn is not homotopy equivalent to Sn for n ≥ 5. 4 Chapter 2 Siegel’s upper half space hg We set some notation and terminology. Throughout this thesis we fix an identification of the standard 2g-dimensional nondegenerate symplectic linear space (R2g , ω) where ω is the 0 −I . A subspace H in 2-form defined by ω(x, y) = t xJy for x, y ∈ R2g and J = I 0 (R2g , ω) is totally isotropic if ω(x, y) = 0 for every x, y ∈ H. If H is totally isotropic, then JH is a subspace transverse to H and the direct sum (H ⊕JH, ω|H⊕JH ) is a nondegenerate symplectic subspace. The maximal dimension of a totally isotropic subspace is seen to be g. A lagrangian subspace is a g-dimensional (maximal) totally isotropic subspace of (R2g , ω). The genus g symplectic group Sp2g R consists of those 2g×2g matrices A satisfying t AJA = J. Equivalently, Sp2g R consists of those linear transformations A satisfying ω(Ax, Ay) = ω(x, y) for all x, y ∈ R2g . One has J ∈ Sp2g R and since every A ∈ Sp2g R satisfies JAJ −1 = t A−1 , the symplectic group is stable under transposition. The linear isomorphism J satisfies J 2 = −id and defines a bilinear form , given by the formula x, y = ω(x, Jy) for x, y ∈ R2g . Evidently , coincides with the standard euclidean inner product on R2g . Trivially J is a , -isometry and ω-symplectomorphism. We call J the standard almost-complex structure on (R2g , ω), and the pair (ω, J) defines the basic linear Kahler structure on R2g . For a subspace H ∈ R2g we denote by H ⊥ , H o the ω, , -orthogonal complements of H respectively. Trivially H ⊥ = (JH)o . The radical of a subspace H ⊂ (R2g , ω) consists of those x ∈ H for which ωx |H ≡ 0. We say vectors x1 , . . . , x2g ∈ (R2g , ω) form a symplectic basis if the matrix x1 . . . x2g ∈ Sp2g R. The standard symplectic basis e1 , . . . , eg , f1 , . . . , fg constitute the columns of the 2g×2g identity matrix I2g . 5 2.1. Symmetric space structure 2.1 2.1.1 Symmetric space structure Cartan decomposition On the real linear symplectic group Sp2g R, the mapping θ : g → t g −1 is called the Cartan involution (evidently θ2 = id). The fixed point set θ(g) = g coincides with the maximal compact subgroup K = Ug in Sp2g R (we have also Ug = Sp2g R ∩ SO2g R). The differential dθ acts on the Lie algebra sp2g R via X → −t X, splitting the Lie algebra into (+1)- and (−1)-eigenspaces k, p. Thus we recover Cartan’s decomposition sp2g R = k ⊕ p. The (+1)-eigenspace coincides with the Lie algebra of the maximal compact Ug . One observes the following relations [k, k] ⊂ k, [k, p] ⊂ p, [p, p] ⊂ k. (2.1) In particular p is not a subalgebra. Explicitly we find p to consist of those matrices of the form A B B −A , where A, B are g × g symmetric real matrices. Cartan’s polar decomposition affirms that the mapping K × p → Sp2g R (K, X) → K · exp(X) is an analytic diffeomorphism. It follows that the homogeneous space hg := Ug \Sp2g R is contractible and that the tangent space at the coset Ke = Ug · e can be identified as TKe hg ≈ p. Notice that since k ∩ p = 0 any subalgebra a of sp2g contained in p is necessarily abelian. The Cartan involution θ : K × P → K × P now assumes the form (k, p) → (k, p−1 ). Let a be a maximal subalgebra of p, necessarily commutative, and set P = exp p, A = exp a. The endomorphisms ad(a) for a ∈ a are simultaneously diagonalizable over R, 6 2.1. Symmetric space structure and sp2g R splits into the centralizer z(a) of a and proper subspaces vα , where α varies over a finite set of nonzero functionals in a∗ (the so-called roots of sp2g R relative to a). By computation one knows [vα , vβ ] ⊂ vα+β , and therefore each vα consists of elements X for which adX is nilpotent (since there are only finitely many nonzero roots). The Cartan involution satisfies dθ : vα → v−α . Choosing an ordering on the roots in a∗ we set n = α>0 vα , and find n to be a nilpotent lie algebra normalized by a. Setting A = expa, N = expn yields closed analytic subgroups of Sp2g R. The Iwasawa decomposition asserts that (k, a, n) → kan is a diffeomorphism K×A×N → Sp2g R. As N is unipotent and A diagonizable over R, the subgroup AN is a connected solvable (triagonalizable) subgroup over R (i.e. the identity component of the R-points of an R-split Borel subgroup). 2.1.2 Killing form and the Ug -invariant metric Recall that for a semisimple lie algebra g, the Killing form B yields a nondegenerate bilinear form defined by B(X, Y ) = trace(ad(X) ◦ ad(Y)), where ad : g → End(g) is the usual adjoint representation given by ad(X) : Z → [X, Z]. For the standard representation of sp2g R (2g ×2g real matrices X satisfying XJ +J t X = 0) the Killing form (up to a constant) takes the form B(X, Y ) = tr(XY ). With respect the Killing form , the decomposition k ⊕ p is orthogonal. Moreover from the standard lie theory [reference] one finds B is negative definite on k, and positive definite on p. We define a right-invariant riemannian metric ·, · on hg by defining X, Y tr(XY ) for all X, Y ∈ p ≈ TKe hg and ·, · Kg = g −1 ·, g −1 · Ke . (2.2) Ke = In other words we ex- tend the Killing form by translation obtaining a Ug -invariant metric on hg for which Sp2g R acts isometrically. 2.1.3 Nonpositive curvature Via the identification TKe hg ≈ p, the riemannian curvature tensor satisfies R(X, Y )Z = −[[X, Y ], Z] and the sectional curvature of a 2-plane spanned by orthonormal vectors Y1 , Y2 in the tangent space Tp hg satisfies K(Y1 ∧ Y2 ) = [Y2 , Y1 ], [Y2 , Y1 ] (c.f. [14]). From the 7 2.1. Symmetric space structure relations (2.1) and (2.2) it follows that K(Y1 ∧ Y2 ) < 0 unless [Y1 , Y2 ] = 0. Thus we find the basic fact that the sectional curvature of a tangent plane is strictly negative unless any two linearly independant tangent vectors have vanishing bracket, i.e. commute. Therefore the abelian subalgebras a of p correspond to tangents of flat subspaces. Evenmore since commuting families of symmetric endomorphisms are simultaneously diagonalizable we a find a direct 1 − 1 correspondance between • flat k-dimensional totally geodesic subspaces in hg ; • k-dimensional abelian subalgebras of p; • k-dimensional R-split algebraic tori in Sp2g R. The maximal dimension of an R-split torus in Sp2g R, the so-called R-rank rankR Sp2g R is known to be g. Indeed, the maximal R-split tori are all conjugate over R to the standard torus −1 diag(λ1 , . . . , λg , λ−1 1 , . . . , λg ) with respect to the standard symplectic splitting e1 , . . . , eg , f1 , . . . , fg . 2.1.4 K¨ ahler structure The vector space p splits into an , -orthogonal direct sum p={ A 0 0 −A }⊕{ 0 B B 0 0 B B 0 }, where A, B are symmetric g × g matrices. The mapping j : p → p defined by j: A 0 0 −A → 0 −A −A 0 , → B 0 0 −B yields an almost-complex structure on p, i.e. j 2 = −id. We extend j to an almost-complex structure j : T hg → T hg by defining jg = g · j ◦ g −1 . Evidently j is pointwise an isometry with respect to the metric , . One finds that j yields a nondegenerate symplectic form Ω on hg defined by Ω(·, ·) = j·, · . Thus hg is a (quasi) K¨ahler manifold with metric , , almost-complex structure j, and compatible symplectic form Ω. Observe that this is consistent with dimR hg = (2g 2 + g) − g 2 being even. 8 2.2. Symplectic lattices 2.1.5 Iwasawa decomposition Every element A ∈ Sp2g R has a unique polar decomposition A = U P , where U ∈ Ug and P is a positive definite symmetric symplectic matrix. A positive definite symmetric matrix P Y 0 I X I 0 is symplectic if and only if P admits a decomposition P = −1 0 Y 0 I X I where X, Y are real g ×g symmetric matrices and Y is positive definite (cf. chapter 5 in [19] ). The X, Y are uniquely determined by P and constitute the so-called Iwasawa coordinates on Ug \Sp2g R. The vector space hg consisting of all pairs Ω = X + iY (i2 = −1) with X, Y g × g symmetric and Y positive definite defines the genus g Siegel upper half space. The Iwasawa coordinates give an explicit identification hg we set 2.2 Ω = X (real part), Ug \Sp2g R. For Ω = X + iY ∈ hg Ω = Y (imaginary part). Symplectic lattices From our point-of-view the Siegel upper half space hg is a Teichm¨ ueller space for 2gdimensional symplectic lattices. Specifically, for A ∈ Sp2g R we say Λ := AZ2g is a symplectic lattice. The symplectic form ω is integral and unimodular on Z2g . Thus ω is integral and unimodular on every symplectic lattice Λ. The unitary group Ug acts simultaneously by isometries and symplectomorphisms on the left. Therefore a choice of representative for A ∈ Ug \Sp2g R determines a well defined isometry and symplectomorphism class of marked symplectic lattice Λ := AZ2g . For any such Λ, a subspace H in (R2g , ω) is Λ-rational if H ∩ Λ is cocompact in H, or equivalently, admits a Z-basis. Given a d-dimensional Λrational subspace H, let x1 , . . . , xd be a Z-basis for H ∩ Λ. Define the volume of H in Λ, denoted vol(H, Λ), to be equal to vol(H/H ∩ Λ). In terms of the Z-basis x1 , . . . , xd we have vol(H, Λ) = + det tx i xj . Since any other Z-basis x1 , . . . , xd of H ∩ Λ differs from x1 , . . . , xd by an integral unimodular substitution, the above formula for vol(H, Λ) is well-defined. We set the volume of the zero subspace in any lattice to be equal to unity. Whenever there is no confusion concerning the ambient lattice Λ we denote the squaredvolume of any Λ-rational subspace H by the shorthand d(H). For vectors x1 , . . . , xd , set d(x1 , . . . , xd ) = d(spanZ {x1 , . . . , xd }). The following inequality is used constantly (cf. [10], [13], [18]): Hadamard inequality. Fix an arbitrary cocompact lattice Γ in RN . Then for any Γ- 9 2.2. Symplectic lattices rational subspaces A, B one has d(A + B)d(A ∩ B) ≤ d(A)d(B). From the identity x, y = ω(x, Jy) and the fact that ω is integral and unimodular on a symplectic lattice Λ in (R2g , ω), it follows that symplectic lattices are self-dual (i.e. isomorphic to their dual lattice Λ∗ which is defined to consist of those vectors v ∈ R2g such that x, y ∈ Z for all y ∈ Λ) with identification given by JΛ = Λ∗ ). The following result of Conway and Sloane (c.f [6], Appendix 2) shows this property in fact characterizes symplectic lattices. Conway-Sloane’s Characterization. A lattice Λ in R2g is symplectic if and only if there is an orthogonal transformation ι on the space such that ι2 = −1 and ι(Λ) is the dual lattice Λ∗ . As corollary we obtain that a so-called classically integral lattice Λ (i.e. Λ∗ = Λ) is symplectic if and only if it is unimodular and has an automorphism (i.e. lattice isometry) ι satisfying ι2 = −1. The result of Conway-Sloane has the merit of showing that many interesting and important lattices are symplectic (after possibly rescaling their volume). Following the nomenclature of [9] we record some instances. All 2-dimensional unimodular lattices are symplectic (this is obviously so, since SL2 = Sp2 ). The extremal lattice D4 on R4 is symplectic whereas the extremal quadratic form E6 in R6 is not self-dual and thus not symplectic. The so-called Barnes-Wall lattices are all symplectic, and moreover (2) + D4m (m ≥ 1), A6 , E8 , K12 , Λ16 and the Leech lattice Λ24 are all symplectic. Their explicit Gram matrices can be found in ([6], Appendix 2). Remark 1. Symplectic lattices also arise naturally from the theory of Riemann surfaces, where the Abel-Jacobi period mapping describes a holomorphic equivariant embedding from the Teichm¨ ueller space of closed hyperbolic genus g surfaces to hg . The problem of characterizing the image of the period mapping, i.e. which symplectic lattices are the period lattices for a riemann surface, is the so-called Schottky problem. 10 Chapter 3 Some symplectic linear algebra With the above preliminaries set, we now describe some symplectic linear algebraic constructions which are basic to our entire paper. 3.1 Intersecting lagrangian subspaces Our first point of study concerns intersections of lagrangian subspaces, where our first lemma is crucial. Lemma 2. Let 1, 2, . . . be lagrangian subspaces in (R2g , ω). Set W = spanR { 1 , 2 , . . .} and K = ∩i=1,2,... i . Then (i) K is the radical of the symplectic subspace (W, ω|W ×W ) (ii) we have an , -orthogonal splitting R2g = W ⊕ JK (iii) the ω-orthogonal complement of K is W , i.e. K ⊥ = W . (iv) dimK + dimW = 2g Proof. If , are distinct lagrangians with x ∈ , then ωx is identically zero on only if also x ∈ if and (by maximality). So for x ∈ W , we have ωx identically zero exactly when x ∈ K. This proves (i). Now JK is trivially , -orthogonal to W exactly because K is the radical of W . Moreover ω is nondegenerate on W ⊕ JK. And since W ⊕ JK contains g-dimensional totally isotropic subspaces (i.e. 1 , 2 , . . .) it follows that W ⊕ JK has dimension at least 2g. The claim in (ii) follows. To establish (iii), we observe that for i = 1, 2 . . . one has K ⊆ i and hence K ⊥ ⊇ ⊥ i = i. So K ⊥ ⊇ W . But both have dimension equal to 2g − dimR K (by (ii)), hence equality. Finally (iv) simply follows from either (iii) or (ii). Our second lemma will play a role in 4. 11 3.2. Heights of ω-orthogonals Lemma 3. Fix a symplectic lattice Λ. Let x1 , . . . , xk be k ≤ g linearly independant lattice elements satisfying: ⊥ xj+1 ∈ x⊥ j ∩ . . . ∩ x1 , j = 1, . . . , k − 1. Then ⊥ (i) for 1 ≤ j ≤ k − 1 the subspace x⊥ j ∩ . . . ∩ x1 is (2g − j)-dimensional, and ⊥ ⊥ 2g (ii) for 2 ≤ j ≤ k − 1 one has the equality x⊥ j + (xj−1 ∩ . . . ∩ x1 ) = R . Proof. For k = 1, (i) is trivial and (ii) is vacuous. Hence we assume k ≥ 2 and proceed by induction on k. For the case k = 2, suppose x2 , x1 are linearly independant and x2 ∈ x⊥ 1. ⊥ Defining |H| := dimR H, the dimension formula from linear algebra yields |x⊥ 2 ∩ x1 | = ⊥ ⊥ ⊥ ⊥ ⊥ |x⊥ 2 |+|x1 |−|x2 +x1 |. Hence (i) and (ii) are equivalent for k = 2, since |x2 | = |x1 | = 2g−1. ⊥ 2g ⊥ ⊥ ⊥ ⊥ ⊥ But x⊥ 1 +x2 = R if and only if either x2 ⊆ x1 or x2 ⊃ x1 . Since the hypotheses x2 ∈ x1 ⊥ ⊥ and x1 ∈ x⊥ 2 are equivalent, we are free to suppose x2 ⊇ x1 . But this containment o ⊥ o is equivalent to having R · Jx2 = (x⊥ 2 ) ⊆ (x1 ) = R · Jx1 , contradicting the linear independance of x2 , x1 . Supposing the lemma has been proven for some 2 ≤ k < g, we’ll now establish the case ⊥ k + 1. The dimension formula shows (i), (ii) are equivalent (since |x⊥ k ∩ . . . ∩ x1 | = 2g − k ⊥ ⊥ ⊥ 2g if and only if x⊥ ⊥ by induction). Now x⊥ k+1 ⊇ (xk ∩ . . . ∩ x1 ) k+1 + (xk ∩ . . . ∩ x1 ) = R (since x⊥ k+1 is a codimension 1 subspace and 2g − 2 ≥ 2g − k). But this final containment ⊥ o ⊥ o ⊥ occurs if and only if Jxk+1 ∈ (x⊥ k ∩ . . . ∩ x1 ) . Now by induction |(xk ∩ . . . ∩ x1 ) | = k. ⊥ o But Jxk , . . . , Jx1 are k linearly independant vectors ∈ (x⊥ k ∩ . . . ∩ x1 ) . Hence we have ⊥ 2g if and only if Jx ⊥ x⊥ k+1 , . . . , Jx1 are linearly dependant, which k+1 + (xk ∩ . . . ∩ x1 ) = R evidently does not occur by the hypotheses of the lemma. This establishes the claim for k + 1, and the lemma. 3.2 Heights of ω-orthogonals The next lemma may properly belong to the category of symplectic geometry-of-numbers, and gives a convenient estimate for the heights of rational lagrangians. 12 3.2. Heights of ω-orthogonals Lemma 4. Let Λ be a symplectic lattice in (R2g , ω) and x ∈ Λ\0. Set Nx = miny∈Λ−x⊥ {|ω(x, y)|}. Then the ω-orthogonal complement x⊥ is a Λ-rational subspace with volume d(x⊥ ) = vol(x⊥ , Λ)2 = ||x||2 Nx−1 . If x is primitive, then d(x⊥ ) = ||x||2 . Proof. We essentially exploit the trivial identity x⊥ = (Jx)o . Indeed the Λ-rationality of x⊥ = (Jx)o is immediate: the almost-complex structure J yields an explicit isomorphism of the symplectic lattice Λ onto its dual Λ∗ = JΛ (ie. symplectic lattices are self-dual), where the equality trivially follows from the fact that ω is integral and unimodular on Λ. The result then follows from §I.5 in [7]. To calculate the volume of x⊥ in Λ we recall the following basic principle in the geometry-of-numbers (cf. §5 in [18]). Let Λ be a 2g-dimensional lattice in the euclidean space (R2g , , ), and H d a d-dimensional Λ-rational subspace with volume vol(H, Λ) = d(H). Set Λ to be the image of the orthogonal projection of Λ onto the (2g−d)-dimensional , -orthogonal complement H o . Then Λ is a lattice in H o with covol(Λ ) = covol(Λ)/d(H). The , -orthogonal complement of x⊥ = (Jx)o is the 1-dimensional subspace spanned by Jx. To determine the image Λ of the orthogonal projection of Λ onto spanR Jx we must describe those possible z = z + λJx ∈ Λ, for which z belongs to x⊥ . But ω(x, z) = λω(x, Jx) must be an integer (since ω is integral on a symplectic lattice Λ). So λ ∈ 1 ω(x,Jx) Z. Note that this explicitly shows Λ is discrete in spanR Jx. Furthermore it is apparent that covol(Λ ) = 1 1 minz∈Λ−x⊥ {|ω(x, z)|} = Nx . 2 ||x|| ||x||2 Since Λ is unimodular we have indeed d(x⊥ ) = ||x||2 Nx−1 , as desired. To see Nx = 1 for x primitive: the quantity Nx is invariant under linear symplectomorphisms, hence we can take Λ = Z2g . That x is primitive means that with respect to the standard Z-basis e1 , . . . , eg , f1 , . . . , fg , one has an expression x = Σi (ai ei + bi fi ), for ai , bi ∈ Z with gcd(a1 , . . . , bg ) = 1. Consequently there exist some ci , di ∈ Z with 1 = Σi (ai ci + bi di ). Set z = Σi (ci fi − di ei ). Then ω(x, z) = 1. 13 Chapter 4 Systoles on hg 4.1 Lagrangian systole function In this section we introduce a systolic invariant of symplectic lattices. This yields an Sp2g Z-equivariant function on hg . It is against this function that we define our equivariant retract. Definition 5. For A ∈ hg let L A denote the set of AZ2g -rational lagrangian subspaces. Take sysL A := min ∈L A vol( , AZ2g ). (4.1) We call sysL the lagrangian systole function. The rational lagrangians L A form a discrete subset of the covolume 1 lattice ∧g Z2g in ∧g R2g . Hence sysL is everywhere finite and nonzero on hg . Moreover the integral symplectic group Sp2g Z acts transitively on L Z2g . Consequently sysL is a Sp2g Z-equivariant function on hg . 4.2 L -wellrounded lattices Before making our next definition, we fix some further Notation 6. For A ∈ hg with Λ = AZ2g a symplectic lattice, • let Lmin Λ denote the set of minimal volume Λ-rational lagrangian subspaces in R2g ; • take WΛ := spanR {Lmin Λ}; • take KΛ = ∩ ∈Lmin Λ ; For a 2g-dimensional symplectic lattice Λ, our earlier Lemma 2 tells us that WΛ = R2g if and only if KΛ = 0. With this equivalence in mind, we now make the central definition of this chapter: 14 4.2. L -wellrounded lattices Definition 7. A symplectic lattice Λ is L -wellrounded if KΛ = 0 (or equivalently WΛ = Λ ⊗ R). We denote by Wg the collection of A ∈ hg for which AZ2g is L -wellrounded. Remark 8. We recall that the 1-systole of an n-dimensional unimodular lattice Γ, denoted sys1 Γ, is defined as sys1 Λ := minx∈Λ−0 ||x||. The 1-systole thus yields an SLn Z-equivariant function on the symmetric space Sn . Alternatively one can define sys1 Γ to be the minimal volume (length) of a Γ-rational 1-dimensional subspace in Rn . Generalizing one obtains the higher dimensional systoles sysk Γ. The classical Hermite constant γn is defined as the supremum of sys1 Γ, where Γ varies over Sn . Minkowski’s first convex body theorem (c.f. [7]) readily impliies γn < ∞, i.e. sys1 is bounded on the noncompact quotient Sn /SLn Z. Minkowski’s convex body theorem yields an estimate which is far from optimal, and whose improvement is an old and central problem within the geometry-of-numbers. One can define the analagous Hermite constant γL ,g := supA∈hg {systL A}. The following proposition shows that γL ,g is always finite. Proposition 9. Fix an integer g ≥ 1. There exists a constant C depending only on g for which every 2g-dimensional symplectic lattice Λ has systL Λ ≤ C. Proof. We describe a procedure using Minkowski’s second theorem on successive minima (cf. §III.2 in [7]) to a construct a rational lagrangian in A ∈ hg whose volume is bounded above by some positive constant depending only on g. The following estimate provided by Lemma 3 will be required: if x1 , . . . , xk (k ≤ g) are linearly independant lattice vectors ⊥ 2g ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ with x2 ∈ x⊥ 1 , x3 ∈ x2 ∩ x1 , . . . , xk ∈ xk−1 ∩ . . . ∩ x1 , then xk + (xk−1 ∩ · · · ∩ x1 ) = R , ⊥ ⊥ i.e. d(x⊥ k + (xk−1 ∩ · · · ∩ x1 )) ≥ 1. As Λ := AZ2g is unimodular, there exists a constant c1 = γ2g depending only on the integer g such that Λ has a nonzero vector x1 of length ||x1 || ≤ c1 . The Λ-rational subspace ⊥ 2 x⊥ 1 has volume d(x1 ) = ||x1 || by Lemma 4. Consider the (2g − 1)-dimensional unit ball B 2g−1 in x⊥ 1 , with euclidean volume σ(2g − 1). Now B 2g−1 is a compact 0-symmetric convex body and hence by Minkowski’s second theorem on successive minima [[7]] we have λ1 λ2 · · · λ2g−1 σ(2g − 1) ≤ 22g−1 ||x1 ||2 , 0 < λ1 ≤ λ2 . . . ≤ λ2g−1 < ∞. ⊥ However x1 ∈ x⊥ 1 , and since x1 is the shortest nonzero lattice point in x1 , we find the first minimum λ1 = ||x1 ||. To estimate the second minimum λ2 , (i.e. the minimal length of a lattice vector x2 ∈ x⊥ 1 ∩ Λ linearly independant from x1 ), we observe that 15 4.2. L -wellrounded lattices λ2g−2 λ2 ≤ λ1 λ2 · · · λ2g−1 and hence 1 λ2 ≤ λ2−2g 22g−1 ||x1 ||2 σ(2g − 1)−1 1 ≤ c2 , where indeed c2 is a constant depending only on g, which explicitly can be taken to be 22g−1 σ(2g − 1)−1 c4−2g . 1 ⊥ ⊥ ⊥ By lemma 3 and Hadamard’s inequality d(x⊥ 2 ∩ x1 ) ≤ d(x1 )d(x2 ) ≤ c2 c1 . We now consider, as before, the (2g − 2)-dimensional unit ball B 2g−2 of euclidean ⊥ volume σ(2g − 2) in the rational subspace x⊥ 2 ∩ x1 . According to Minkowski’s second theorem the new successive minima λ1 , λ2 , . . . , λ2g−2 (which now depend on the unit ball ⊥ B 2g−2 in x⊥ 2 ∩ x1 ) satisfy λ1 λ2 · · · λ2g−2 σ(2g − 2) ≤ 22g−2 ||x2 ||2 ||x1 ||2 . ⊥ As indeed x1 , x2 are the 1st, 2nd shortest nonzero lattice vectors in x⊥ 2 ∩ x1 , we find that λ1 = ||x1 ||, λ2 = ||x2 ||. Consequently, we can estimate λ3 (the minimal length of a lattice vector x3 linearly independant from x2 , x1 such that z, x2 , x1 span a totally isotropic subspace) as before: 4−2g 2g−2 2 ||x1 ||2 ||x2 ||2 σ(2g − 2)−1 λ3 ≤ λ−1 1 λ2 ≤ c3 , where c3 is again some constant depending only on g. ⊥ ⊥ Continuing this process for x⊥ 3 ∩x2 ∩x1 , etc we find linearly independant lattice vectors ⊥ xg , . . . , x1 spanning a rational lagrangian subspace L := x⊥ g ∩ . . . ∩ x1 with ||xk || equal to ⊥ the k th successive minima of the (2g − k + 1)-unit ball in x⊥ k−1 ∩ · · · ∩ x1 . The Hadamard inequality gives d(L) ≤ 1≤i≤g ||xi ||2 ≤ cg , where cg will be a positive contant depending only on g. Remark 10. Siegel’s upper half space hg admits the following interpretation. For a given symplectic manifold (M, ω) an endomorphism J : T M → T M of the tangent bundle (fibrewise linear) satisfying J 2 = −id is known as an almost-complex structure. We say an almost-complex structure J is ω-compatible if the symmetric form gJ (·, ·) := ω(·, J·) is 16 4.2. L -wellrounded lattices a riemannian metric. The collection Jc (ω) of ω-compatible almost-complex structures J on (M, ω) is a contractible space. Consider the 2g-dimensional torus T 2g endowed with a unimodular and integral translationinvariant symplectic form ω (i.e. R2g /Z2g ). A marking A ∈ hg yields a ω-compatible 0 −I . The metrics obtained almost-complex structure J := AJstd A−1 , where Jstd := I 0 are flat metrics on T 2g . Thus we find that hg can be identified with the collection Jcf lat (ω) of flat ω-compatible almost-complex structures on (T 2g , ω). Our lagrangian systole function sysL thus extends to a function defined on the (infinitedimensional) space Jc (ω). The question of whether or not supJ∈Jc (ω) sysL < ∞ is unknown. This author sees the present thesis as marking preliminary progress towards answering the previous question, i.e. does sysL satisfy a systolic-inequality on Jc (ω), and hopes to address these issues in the future. 17 Chapter 5 The equivariant retract Wg In this chapter we make precise the continuity of our deformation of hg onto the set of L -wellrounded symplectic lattices Wg . Proposition 11. For g ≥ 1, we have a strong Sp2g Z-equivariant deformation retract hg → Wg . We begin with a preliminary Definition 12. For j = g, g + 1, . . . , 2g − 1 let h(j) g := {A ∈ hg |dimR (spanR Lmin A) ≥ j}. (j) The subspaces hg are Sp2g Z-equivariant and yield the following filtration of the genus g Siegel upper half space (g+1) hg = h(g) ⊇ · · · ⊇ h(2g) = Wg . g ⊇ hg g The inclusions are not generally strict. Nonetheless, we still have our next (j+1) Proposition 13. For j = g, . . . , 2g − 1, the space hg (j) is a closed subset of hg . Proof. It is here convenient to view hg as a space of certain flat metrics on Z2g and R2g : for A ∈ hg we define a metric gA on both Z2g , R2g by setting gA (x, y) := ω(Ax, JAy). The metric gA is independant of choice of representative of A ∈ Sp2g R since any U ∈ Ug is a symplectomorphism with t U = U −1 . (j+1) Suppose now (Ak ) is a sequence in hg with limit A∞ in hg . Take gk := gAk and g∞ := gA∞ . Then (gk ) is a sequence of flat metrics for which the set Lmin gk of minimal gk volume Z2g -rational lagrangians span a real subspace of dimension ≥ j+1. As the collection of Z2g -rational lagrangians is discrete, by passing to a subsequence we can assume that the set Lmin gk is constant as k → ∞. Therefore the real span of Lmin gk is some constant 18 Chapter 5. The equivariant retract Wg subspace, say W . Hence the real span of Lmin g∞ is also W . Therefore g∞ and A∞ belong (j+1) to hg . (j) We will construct our retract hg → Wg from a sequence of retractions hg = hg (j+1) hg → ... → (2g) hg → = Wg . The following proposition constructs these auxiliary retrac- tions. (j+1) Proposition 14. For j = g, . . . , 2g−1, the space hg retract of is a Sp2g Z-equivariant deformation (j) hg . Proof. To construct a retraction, we first describe some 1-parameter families of linear symplectomorphisms. For K a totally isotropic subspace of (R2g , ω, J), take a , -orthocomplement V to K in K ⊥ . Now for each µ ∈ R consider the map TµK : K ⊕ V ⊕ JK → K ⊕ V ⊕ JK defined as k + v + x → µk + v + µ−1 x (ie. TµK = µ ⊕ id ⊕ µ−1 ). Hence TµK dilates radially by µ on K, is identity on the orthocomplement V , and contracts by µ−1 on JK. Now we claim TµK ∈ Sp2g R for µ ∈ R whenever K is totally isotropic. Indeed K, JK span a J-invariant subspace with ω-orthogonal (equivalently, , -orthogonal) V , so ω(V, JK) = ω(K, V ) = 0. Thus TµK can directly be seen to preserve ω. The definition of TµK is independant of choice of , -orthocomplement V in K ⊥ . (j) Consider the action of TµKA on a marking A ∈ hg , where we expand along the total intersection KA of the minimal volume rational lagrangian subspaces in Λ = AZ2g . That is, consider the half-ray TµA A := TµKA A in hg where µ ≥ 1. The linear symplectomorphisms TµA = TµKA have the effect of expanding the volumes of the rational minimal lagrangians Lmin A proportionally by a factor of µd/2 (where d = dimKA = 2g−j −1) while contracting in the direction of JK. Now there shall exist a minimal time τ = τ (j) A ≥ 1 for which (j+1) TτA A ∈ hg , ie. for which Lmin TµA A spans a linear subspace of dimension ≥ j + 1. (j+1) Evidently τ (j) is Sp2g Z-equivariant and τ (j) A = 1 for A ∈ hg We claim the function τ (j) is continuous on (j) hg (j) hg . . Indeed suppose (Ak ) is a sequence in with Ak → A∞ . We want to show τ (j) Ak → τ (j) A∞ . (j+1) If A∞ ∈ hg (5.1) (j+1) , then for sufficiently large k we’ll have Ak ∈ hg Suppose now instead that A∞ ∈ to WA∞ , KA∞ in R2g . (j) (j+1) hg \ hg . and (5) will be clear. Then WAk , KAk Gromov-Hausdorff converge Consequently for sufficiently large k, both the lattices Ak Z2g , A∞ Z2g and the deformations TµAk , TµA∞ will be comparably close. So the times τ (j) Ak , τ (j) A∞ 19 Chapter 5. The equivariant retract Wg for which the corresponding deformations generate a “new” minimal lagrangian will be comparably close. That is, τ (j) Ak → τ (j) A∞ . (j) The continuity of the function A → τ (j) A on hg implies that (j) A r(j) : [0, 1] × h(j) g → hg , (t, A) → T(1−t)+tτ (j) (A) A is a continuous homotopy. It is evidently Sp2g Z-equivariant. We now complete the proof of Proposition 11: Proof of Prop 11. The successive homotopies r(g) , r(g+1) , . . . , r(2g−1) constructed above yield an equivariant strong deformation retract hg → Wg . 20 Chapter 6 Cocompactness of Wg In this section we prove that our closed Sp2g Z-equivariant deformation retract Wg is cocompact, ie. the quotient Wg /Sp2g Z is compact in hg /Sp2g Z. To verify the cocompactness we use a variation on the well-known Mahler compactness criterion. To present this criterion we let Pg denote the vector space of g × g real symmetric positive definite matrices and recall the natural right action of GLg Z on Pg given by (g, P ) → t gP g. Mahler criterion. A closed subset K ⊆ Pg /GLg Z is compact if and only if the following two conditions are satisfied: (i) there exists C > 1 such that C −1 ≤ |detg Y | ≤ C for every Y ∈ K (where detg is the g-dimensional determinant function on Pg ); (ii) there exists (g) > 0 such that syst1 Y ≥ (g) for every Y ∈ K (where syst1 is the g-dimensional 1-systole function on Pg ). The above criterion is proven by examining the cusps in a fundamental domain M of GLg Z in Pg (cf. [11], [19]). We shall likewise identify a cocompactness criterion for closed subsets of hg by considering a fundamental domain for the integral symplectic group first constructed by C.L. Siegel in [19]. Proposition 15 (Siegel). Let F be that subset of hg which in the Iwasawa coordinates consists of those Ω = X + iY for which the following three conditions are satisfied: (i) |xij | ≤ 1/2, for X = Ω = (xij ); (ii) |detY ∗ | ≤ |detY | for all Y ∗ = (Ω.γ), where γ ∈ Sp2g Z and Y = Ω; (iii) Y ∈ M Since the cube |xij | ≤ 1/2 is compact, Siegel’s fundamental domain F and the Mahler criterion for Pg yields the following 21 Chapter 6. Cocompactness of Wg Mahler-Siegel criterion. A closed subset K ⊆ hg /Sp2g Z is compact if the following conditions are satisfied: (i) detg is bounded on (ii) there exists K; (g) > 0 such that syst1 Y ≥ for all Y ∈ K. (2g) To apply Prop 6 we shall require systL and syst1 to both be bounded away from zero. This is established by the following (i) If A ∈ W , then 1 ≤ covol(spanZ Lmin A) ≤ (systL A)2 ; Proposition 16. (ii) there exists > 0 such that if A ∈ hg admits a nonzero lattice vector x with ||x|| < , then Lmin A ⊂ x⊥ . (iii) there exists for all A ∈ W > 0 such that syst1 A ≥ Proof. Set Λ = AZ2g and define δmin = systL Λ. Let Λ-rational lagrangians. We set 1...k = 0 and 1···ˆi···k ij··· = i ∩ j 1, . . . , k be distinct minimal volume ∩ · · · . We choose k minimal such that = 0 for 1 ≤ i ≤ k. The generalized Hadamard inequality 2.2 yields the estimates 1 ≤ d( 1 + ··· + k) ≤ δmin d( 2 + · · · + d( 1 ∩ ( 2 + · · · + k ≤ δmin k) k )) d( i ∩ ( i+1 + ··· + k )). 1≤i≤k−1 We claim that for j = 1, . . . , k − 1 the subspace j+1,...,k + Indeed by Lemma 2 we see j ∩( j+1 j+1,...,k + ··· + k) is a rational lagrangian. is the radical of ωx vanishes identically on the totally isotropic subspace x∈ j+1 + · · · + k . j j+1 + ··· + ∩( j+1 k; + ··· + (6.1) in particular k) for every So (6) is totally isotropic. From Lemma 3 and the standard dimension formula from linear algebra it follows that (6) is g-dimensional, hence lagrangian. Consequently d( j+1,...,k +. j ∩( j+1 +···+ k )) ≥ δmin . Together with the Hadamard inequality we get δmin d( j,j+1,...,k ) ≤ d( d( j+1,...,k) j ∩( j+1 + ··· + k )). 22 Chapter 6. Cocompactness of Wg and altogether we find 1 ≤ d( 1 + ··· + k) k−1 k ≤ δmin δmin j=1,...,k−1 d( j...k ) 2 = δmin = (systL Λ)2 . d( j+1,...,k ) This proves (i). To prove (ii) we show = 1 suffices. Take A ∈ hg and set Λ = AZ2g . Let x be a primitive nonzero lattice vector in Λ and suppose or equivalently x ∈ / . Then is a Λ-rational lagrangian with x⊥ , ∩ x⊥ is a (g − 1)-dimensional Λ-rational totally isotropic subspace. Since + x⊥ = R2g , the Hadamard inequality and Lemma 4 yields d( ∩ x⊥ ) ≤ d( )d(x⊥ ) = d( )||x||2 . Let z1 , . . . , zg−1 be a Z-basis for ( ∩ x⊥ ) ∩ Λ. Now x, z1 , . . . , zg−1 ∩ x⊥ ). Therefore are linearly independant (since x ∈ / Λ-rational lagrangian contained in x⊥ . := spanR {x, z1 , . . . , zg−1 } is a From the Hadamard inequality we find d( ) ≤ ||x||d(z1 , . . . , zg−1 ) ≤ ||x||3 d( ). So if ||x|| < 1, then d( ) < d( ). Therefore any symplectic lattice Λ admitting a nonzero lattice element x with ||x|| < 1 has Lmin Λ ⊂ x⊥ . This proves (ii). Taking = 1 one deduces (iii) immediately from (ii). Finally we have Proposition 17. (i) detg is bounded on (ii) there exists > 0 such that syst1 Y ≥ (g) W and for all Y ∈ (W ∩ F). Proof. We establish (i) by showing separately detg is bounded below and above on (W ∩F). First a trivial computation: a point Ω ∈ hg with Iwasawa coordinates X + iY uniquely Y YX determines a positive definite symmetric symplectic matrix A = . XY XY X + Y −1 Consequently the first g columns Y of A generate a Λ-rational lagrangian subspace, XY say A , with Λ = AZ2g . But one readily computes vol( A , Λ) = |detY | |det(Ig + X 2 )|, where Ig is the g × g identity matrix. For A (or equivalently Ω) in W , Prop 16 gives a lower bound |det A| ≥ |det(I + A2 )|−1/2 . 23 Chapter 6. Cocompactness of Wg A runs over a compact set as A varies over F ∩ W , we find |det(I + Since bounded image in R>0 . This finds |detg | is bounded from below on To find an upper bound for |detg | on Λ= AZ2g is self-dual, ie. Λ and Λ∗ A2 )| has (W ∩ F). W , we argue as follows. A symplectic lattice are isometric. Explicitly one can take Λ∗ = JΛ. Since A ∈ Sp2g R if and only if t JAJ = t A−1 , we can explicitly take Λ∗ = JΛ = t A−1 Z2g . Let us also make the observation that W ∗ = W , ie. Λ ∈ W if and only if Λ∗ ∈ W . This is obvious because J is simultaneously an isometry and symplectomorphism. If A ∈ W ⊂ hg has Iwasawa coordinates X + iY , then t A−1 = A−1 has Iwasawa coordinates (−X) + iY −1 . Hence we find, as before, that 1 ≤ |detY |−1 This establishes that |detg | is also bounded from above on |det(I + X 2 )|. W . This proves (i). We establish (ii) by contradiction. Suppose we had a sequence (Ak ) in W ∩ F with (g) syst1 Yk → 0, where Yk := Ak Z2g Ak and Xk := Ak . Set Γk to be the rank g subgroup of Yk . generated by the g columns of Xk Yk (2g) One obviously has syst1 (2g) Ak ≤ syst1 Γk . Choose yk ∈ Yk Zg to be a 1-systole, i.e. (g) syst1 Yk . Then there exists some nonzero wk in Zg for which yk = Yk · wk . The Yk yk · wk = is a Γk lattice vector, with vk = Xk Yk · wk = Xk yk . vector y˜k := Xk Yk vk By the triangle inequality one finds ||yk || = (g) ||y˜k || ≤ ||yk || + ||vk || = syst1 Yk + ||Xk yk || (g) ≤ syst1 Yk (1 + ||Xk ||∞ ), where || · ||∞ denotes the usual operator sup-norm. Since the Xk vary over a compact set, the term 1 + ||Xk ||∞ is bounded, and we (g) find syst1 (1 + ||Xk ||∞ ) → 0 as k → ∞. But this means the sequence (Ak ) ∈ W has (2g) syst1 (g) Ak → 0, which contradicts (iii) of the previous proposition 16. Therefore syst1 is bounded away from zero on (W ∩ F). The above proposition together with 6 yields Proposition 18. For g ≥ 1, the retract Wg is cocompact in hg . 24 Chapter 7 Codimension of Wg Let us begin by describing the maximal R-split tori in Sp2g R. We shall be rather formal throughout this section because we are looking to derive the precise formula 7.1. From the theory of reductive algebraic groups one knows that all maximal R-split tori are conjugate in Sp2g R. With respect to the standard symplectic basis e1 , . . . , eg , f1 , . . . , fg the standard maximal R-split torus has the form −1 ∆ = diag(λ1 , . . . , λg , λ−1 1 , . . . , λg ). We prefer the following interpretation of ∆. For any anisotropic pair of vectors x, y ∈ R2g (i.e. ω(x, y) = 0) one obtains an ω-orthogonal splitting R2g = R(x, y) ⊕ R(x, y)⊥ . The symplectic subspace R(x, y) thus supports an embedding SL2 R → Sp2g R, where SL2 R acts faithfully on R(x, y) and trivially on the orthogonal complement R(x, y)⊥ . With respect to the given basis x, y for R(x, y) we determine a rank 1 R-split torus represented by ψ{x,y} : λ → λ 0 0 λ−1 . Likewise any ω-orthogonal splitting R2g = R(x1 , y1 ) ⊕ . . . ⊕ R(xg , yg ) 25 Chapter 7. Codimension of Wg yields a g-dimensional R-split torus generated by the commuting elements ψ1 , . . . , ψg , ψi := ψxi ,yi . Since Sp2g R acts transitively on symplectic bases for R2g one finds that indeed all maximal R-split tori are obtained in this fashion. A choice of symplectic basis F := {xi , yi }i=1,...,g moreover allows us to explicitly coordinatize the connected component of the above maximal R-split torus ∆F as Rg>0 by first identifying its Lie algebra with Rg (via roots) and using the exponential map. More simply this amounts to the following isomorphism κ = (κ1 , . . . , κg ) → ψi (exp κi ), κ ∈ Rg . i The euclidean structure , on R2g induces an inner product structure on the g th exterior power ∧g R2g . On simple g-vectors x = x1 ∧ · · · ∧ xg , the inner product is given by x, x = det( xi , xj ). With this metric structure the volume of the fundamental parallelipiped spanned by vectors x1 , . . . , xg in R2g corresponds to the length of x1 ∧ · · · ∧ xg in ∧g R2g . For a symplectic lattice Λ, rational lagrangian , and maximal torus ∆F we should like an expression for the differential of vol( , φκ Λ) seen as a function of κ on the Lie algebra of ∆F . We proceed with a very elementary computation. A symplectic basis {xi , yi } giving the frame F yields a basis for ∧g R2g , with respect to which the squared length of an arbitrary simple g-vector ξ = x1 ∧ · · · ∧ xg can be computed as ||ξ||2 = ||projη ξ||2 , η where η runs over the basis induced by F , and the vector projη ξ denotes the orthogonal projection of ξ onto the line Rη. For a basis element η we have ψκ η = exp( where j (η) j j (η))η, is some integer in {0, ±1, ±2} depending on η. Thus we find the fomula ||ψκ ξ||2 = e j j (η)κj ||projη ξ||2 . η Now for a fixed g-vector ξ we see the expression ||ψκ ξ||2 as a function on the Lie algebra 26 Chapter 7. Codimension of Wg of ∆F in the variable κ = (κ1 , . . . , κg ). With respect to the exterior differentiation d = ∂ ∂κi dκi we find the formula d||ψκ ξ||2 = i (η) i ·e 1 (η)κ1 +···+ g (η)κg · ||projη ξ||2 dκi . η Evaluating the differential at κ = (0, . . . , 0) yields d||ψκ ξ||2 |κ=0 = 2 i (η)||projη ξ|| i Suppose now 1, . . . , k dκi |κ=0 . (7.1) η were exactly the minimal volume rational lagrangians in a symplec- tic lattice Λ, hence having volumes all pairwise equal. Suppose ξ1 , . . . , ξk were g-vectors representing each 1, . . . , k . For a given maximal R-split torus, say ∆F , the problem of finding a 1-parameter subgroup {φt } with φ0 = id in ∆F with the property that vol( 1 , φt Λ) = · · · = vol( k , φt Λ) for all t ≥ 0 therefore reduces to the problem of finding a 1-dimensional subspace on which the linear forms d||φt ξ1 ||2 |κ=0 , . . . , d||φt ξk ||2 |κ=0 coincide. This however is simply a question of linear algebra. In particular, we find that if the linear forms {d||φt ξi ||2 |κ=0 }i=1,...,k are linearly independant (and in particular with k ≤ g), then there will always exist a one-dimensional subspace within the Lie algebra of ∆F on which they agree. That is, there will exist a 1-parameter subgroup in ∆F which will deform their volumes equally. A sufficiently small identity neighborhood of this 1-parameter subgroup will generate a 1-dimensional variation of Λ on Wg . As corollary to this discussion we find Corollary 19. The set of L -wellrounded lattices Wg has codimension 1 in hg . More precisely, the codimension of a cell containing a symplectic lattice Λ ∈ Wg which contains exactly k lagrangian systoles will be < k. If the corresponding lagrangian systoles are linearly independant in the sense that the associated linear forms are linearly independant, then the codimension is precisely k − 1. Therefore we properly deduce that Wg contains a codimension 1 cell by exhibiting a symplectic lattice which admits exactly two transverse lagrangian systoles. It will suffice to give an example in dimension 4. Namely 27 Chapter 7. Codimension of Wg consider the matrix λ λx 0 −1 0 λ 0 0 0 λ−1 0 0 −λx 0 0 , 0 λ where λ > 1 and x is a small strictly positive number. The images of the standard lagrangians (e1 , e2 ), (f1 , f2 ) are exactly the lagrangian 1-systoles. Evidently the parameter λ yields a 1-parameter subgroup of deformations tangent to Wg . Remark 20. The previous computation justifies the heuristic that having two lagrangians of equal volume is a codimension one condition on the symmetric space hg . The author was reluctant to accept this cavalier approach. Such reasoning is often employed in justifying that the wellrounded retract Xn of the symmetric space Sn has codimension n − 1, there the heuristic being that n vectors having equal length amounts to (n − 1) equations and thus ‘wellrounded-ness’ is a codimension n − 1 condition. We can however justify this reasoning, as above. Finding a 1-parameter subgroup of some maximal R-split torus in SLn R which grows k equal length linearly independant lattice vectors amounts to finding a 1-dimensional subspace within the Lie algebra of the torus on which corresponding k linear forms coincide. By linear algebra we of course know that one can manage to find a codimension k − 1 linear subspace. 28 Chapter 8 Bavard’s retract Yg Our set Wg of L -wellrounded lattices is not the first codimension 1 Sp2g Z-equivariant strong deformation retract of hg constructed. Extending the wellrounded retract of Sn Bavard [3] constructed a rather obvious codimension 1 retract. It is convenient for the discussion to introduce the following Notation 21. For a symplectic lattice Λ, set S1 Λ := spanR sys1 Λ. Bavard’s retract is the subset Yg of hg consisting of those 2g-dimensional symplectic lattices Λ for which S1 Λ is anisotropic, i.e. there exist distinct x, y ∈ sys1 Λ for which ω(x, y) = 0. The deformation of hg onto Yg is motivated by the wellrounded retract on Sn , and can be described as simply ‘push the wellrounded retract until one can’t’. For given Λ ∈ hg , if S1 Λ is anisotropic then we do nothing and leave Λ stationary. Otherwise S1 Λ is totally isotropic and , -orthogonal to JS1 Λ, thus we find a splitting R2g = S1 Λ ⊕ JS1 Λ ⊕ Q, where Q is an , -orthocomplement. Now we define the deformation at Λ to expand radially on S1 Λ, and compensate by contracting radially on JS1 Λ. This yields a symplectic deformation stabilizing the above , -orthogonal splitting. Evidently there will exist a minimal time at which a ‘new’ 1-systole, which necessarily does not belong to S1 Λ, will appear. At this point either the new 1-systoles span an anisotropic subspace (and we stop), or we deform within the new larger totally isotropic S1 Λ . Thus we find the continuous Sp2g Z-equivariant strong deformation retract Yg . If Λ ∈ Yg , then there exist 1-systoles x, y with ω(x, y) ∈ Z − {0}. From the Wirtinger 29 Chapter 8. Bavard’s retract Yg inequality (i.e. Cauchy-Schwartz) we have 1 ≤ ω(x, y) ≤ ||x|| · ||y|| = sys1 Λ2 . Thus the 1-systole function sys1 is bounded from below on Yg . From our previous arguments on the cocompactness of Wg it follows that Yg is cocompact in hg (compare with [3], Theorem 2). One readily sees that Bavard’s retract Yg possesses cells of codimension 1 in hg . Consider the ω-orthogonal symplectic splitting Z4 = Z(e1 , f1 ) ⊕ (e2 , f2 ). Contracting the angle between the perpendicular vectors e2 , f2 , say via transvections along the diagonal e2 + f2 , i.e. deforming via the matrix Aλ := 1 1 + 2λ 1+λ λ λ 1+λ grows the lengths of both e2 , f2 . Thus the symplectomorphism id ⊕ Aλ : R(e1 , f1 ) ⊕ R(e2 , f2 ) → R(e1 , f1 ) ⊕ R(e2 , f2 ) describes a 1-parameter family (with λ > 0 sufficiently small) of 4-dimensional symplectic lattices having exactly two 1-systoles (namely e1 , f1 ), and which have the property of spanning an anisotropic subspace. Remark 22. The intersection between our retracts Wg , Yg is unclear. Moreover we cannot say `a priori which retract (if any) posses a further codimension one equivariant retract, i.e. can the retracts be pushed further? 30 Chapter 9 Weak and strong spines Our initial notion of spine is perhaps unreasonably strong. To illustrate, consider how following problem of the weak spine essentially differs from its original statement in (1.1). Problem of the Weak Spine. For every finite-index torsion free subgroup Γ of Sp2g Z, construct a codimension g cocompact Γ -equivariant strong deformation retract W = WΓ of hg . Recall that the torsion free finite-index subgroups Γ act freely on hg , i.e. all point stabilizers are trivial, and therefore any retract WΓ having the above properties yields a compact manifold WΓ having the minimal possible (geometric) dimension. Observe that constructing an Sp2g -equivariant spine W yields a weak spine, since the natural projection W /Γ → W /Sp2g Z is a finite topological covering map for every finite index subgroup Γ ≤ Sp2g Z. From the point of view of group cohomology the question of constructing a weak spine is perhaps more natural, especially considering the following fact (c.f. [5] VIII.7.2). Proposition 23. If Γ is any finite index torsion-free subgroup of Sp2g Z (g ≥ 2) with cd Γ = d then there exists a d-dimensional finite CW-complex Y realizing the EilenbergMaclane classifying space K(Γ , 1). Note that from [4] we know d = g. The conclusion means that Γ acts freely on some d-dimensional contractible finite CW complex Y . As stated in [5] the proposition has the form “geometric dimension of Γ coincides with its cohomological dimension”, and its proof shows this to be definitely true so long as d ≥ 3. The model space Y constructed is however just an abstract CW complex, which a priori might not even admit an embedding into any small-dimensional euclidean space. A ‘procedure’ for how one should obtain such a Y given some larger dimensional K(Γ , 1) model (e.g. hg ) on which Γ already acts freely is due to Stallings [21]. 31 Chapter 9. Weak and strong spines Proposition 24. [Stallings] Given a k-dimensional polyhedron K, a d-dimensional manifold M , and a homotopy equivalence f : K → M , there exists a procedure which, whenever k ≤ m − 3, yields a k-dimensional subpolyhedron K1 ⊂ M and a simple homotopy equivalence K → K1 for which the composition K → K1 ⊂ M is homotopic to f : K → K1 . Stallings’ method belongs to the PL-topology, wherein a simple homotopy equivalence refers to a finite sequence of elementary cellular collapses and expansions (c.f. [17]). Let us remark that the existence of initial homotopy equivalences f : Y → hg follows from the fact that both Y, hg are models for K(Γ , 1), and can easily be constructed (given a reasonable description of Y ). The procedure given by Stallings coupled with 23 at least guarantees the existence of a cellular weak spine for every finite index torsion-free subgroup Γ . However this abstract proof of existence is still well short of yielding any concrete retract, and therefore tremendously unsatisfactory. A more refined investigation into the existence of strong spines is given by [8], wherein a cohomological characterization is given for, e.g. those arithmetic groups Γ which act properly on some contractible CW-complex having dimension equal to vcd Γ (c.f. [8], Theorem III]). As particular application the authors conclude that a strong spine for Sp4 Z exists. However no explicit model is given, nor any guarantee that this model can be realized as equivariant retract of h2 . On the other hand, weak spines for h2 were exhibited in [15] and the argument will be discussed in chapter10 of this thesis. However it can be seen that their method does not yield an Sp4 Z-equivariant retract. 32 Chapter 10 MacPherson and McConnell’s explicit reduction theory This chapter describes MacPherson and McConnell’s construction [15] of a weak spine for Sp4 R. Our discussion emphasizes two aspects. Firstly, their retraction derives from a fortuitous regular cell structure on the smooth quotients h2 /Γ, for Γ a finite index torsion free subgroup of Sp4 Z. This cell structure is obtained only after elaborate and exhausting computations. Secondly, their method does not yield a regular cell structure for the whole arithmetic subgroup Sp4 Z - and consequently does not yield a retract of h2 /Sp4 Z. Thus we have explicit weak spines without a strong one. Besides their original paper [15], we have found Gunnell’s thesis [12] useful in understanding MacPherson and McConnell’s construction. 10.1 Voronoi’s cell decomposition There is a confusion throughout the literature as to the meaning of Voronoi cells. This possibly arises from a forgotten fact that Georges Voronoi wrote two large memoirs (cited as V1, V2 in [15]) in which he described two distinct reduction theories for quadratic forms. The second reduction theory (wherein the well-known Delauney-Voronoi tessalations arise) is commonly used in describing lattice packings (e.g. [9]). We must however refer to the lesser-known reduction theory, one in which the notions of eutaxy and perfection are key (c.f. [13], [1]). The reader will find here no application of these terms (which are central to Voronoi’s theory) for the simple reason that the author finds them abominable. In fact, we shall develop only enough of Voronoi’s theory to indicate the form of MacPherson and McConnell’s weak spine. Our first step will be to recognize the space Pn of symmetric positive definite n × n matrices as a linear symmetric space. We establish some terminology. For any nonzero vector v ∈ Rn+1 we call the set R>0 v of 33 10.1. Voronoi’s cell decomposition nonzero positive scalar multiples of the vector v a ray. A cone is a union of rays. Suppose C is a convex cone in Rn+1 containing no line through the origin and with nonempty interior. Then the projectivization PC, i.e. the topologized set of rays in C, is homeomorphic to an n-ball D. Consider a symmetric space M = K\G (where, as usual, K is a maximal compact of a semisimple Lie group G) endowed with the following properties: 1. There is a homeomorphism φ : M → interior(D) between M and the interior of the ball D, and 2. there is a group homorphism i : G → GL(Rn+1 ) making φ equivariant. So for each automorphism g of M the linear transformation i(g) preserves the cone C. A symmetric space admitting such a structure is said to be linear. An essential aspect concerning linear symmetric spaces is how easily one obtains an equivariant compactification. The closure C of a cone C in Rn+1 permits us to take the closure D of D in C. With respect to our mapping φ, we see D as equivariantly compactifying M . A second notable feature of linear symmetric spaces is that convex hulls of subsets are well defined. The first claim we want to justify is the following Proposition 25. The symmetric space Pn is linear. Proof. The space Pn sits as a convex cone in RN +1 := Rn(n+1)/2 . From the polar decomposition for GLn (R) it follows that Pn is homeomorphic to the symmetric space SOg R\GLn (R)o of positive-definite quadratic forms on Rn (equivalently, marked oriented n-dimensional cocompact lattices in Rn ). The closure Pn+ of the cone Pn in RN +1 consists of n×n symmetric semi -definite matrices. The projectivization of Pn is seen to coincide with our symmetric space Sn of unimodular positive-definite quadratic forms. In other words Sn is obtained via a section of the cone (i.e. principal R>0 -bundle) Pn . We remark that an explicit homeomorphism of Sn with the unit disk in RN +1 is given by the so-called Cayley transform (e.g. Theorem 6.3.1 in [19]). In dimension 2, identifying S2 with the upper half plane H ⊂ C the Cayley transform has the form z → (z −i)(z +i)−1 . As described above, the equivariant homeomorphism between Sn and the interior of the N -ball yields an equivariant compactification Sn . Now we will describe a cell decomposition of Sn . This arises from first taking convex hulls of points on the boundary ∂Pn+ which consists of semi-definite quadratic forms. One easily generates semi-definite forms by using 34 10.1. Voronoi’s cell decomposition the dual lattice Zn∗ which consists of those linear functionals λ ∈ Rn∗ for which λ(x) ∈ Z for all x ∈ Zn . For any (nonempty) finite set X = {λ1 , . . . , λk } ⊂ Zn∗ set k ρi λ2i for ρi > 0}. C(X) = {θ ∈ Pn : θ = i=1 The elements λ21 , . . . , λ2k are semi-definite forms lying on the boundary of the closure Pn+ in Pn , and one finds C(X) to be the relative interior of a convex cone in Pn . Let us now n∗ consisting of primitive linear functionals, i.e. those λ such specify the subset Zn∗ prim of Z that λ(v) = 1 for some v ∈ Zn . Let R := C(Zn∗ prim ) denote the convex hull of the functionals (lv )v∈Zn . The convex hull R has codimension 1 in Pn and the structure of an infinite polyhedron. The facets of R are certain convex polyhedrons with vertices lv1 , . . . , lvk . Projecting R onto the compactified section Sn yields an GLn Z-equivariant polyhedral decomposition of Sn . The restriction of this polyhedral structure to the interior Sn is Voronoi’s cell decomposition. We remind the reader that Σ is said to be a finite regular cell complex structure on the topological space X if we have a finite collection of closed balls σ ∈ Σ with X = ∪Σ σ whose interiors intσ form a setwise partition of ∆ and such that for any σ ∈ Σ the boundary ∂σ is a union of members of Σ. The main theorem of Voronoi’s reduction theory states that the above projection yields a SLn Z-equivariant locally finite regular CW-complex structure on Sn . To this point we have been terse in our descriptions and are unable to precisely illustrate the facets of the polyhedral structure on Sn . We refer the reader to §II.3-4 in [12] for details. In particular, one finds Voronoi’s cell decomposition on S2 (identified with the upper half plane H) to be given by Figures II.5 in [12]. It is significant to our discussion to note that Voronoi’s cell structure on H is different than that given by the usual fundamental domain of SL2 Z (e.g. [5] VIII.9.2). 35 10.2. Embedding h2 into S4 10.2 Embedding h2 into S4 We must first be clear that Siegel’s upper half space h2 is not linear. It can however be embedded as a convex subset of S4 . Indeed the inclusion Sp4 R → SL4 R yields an equivariant totally geodesic embedding h2 → S4 . Let M denote the image of h2 with respect to this embedding. The linear structure on P4 gives an equivariant compactification M of M by taking the relative closure of M in S4 . The intersections of M with the Voronoi cells σ of S2 gives a decomposition of M into sets(!) M ∩ σ. Now let us vigorously stress that a priori one is very far from being able to claim that these M ∩ σ yield a cellular decomposition (as defined in 10.1). Indeed one does not know beforehand which intersections are nonempty, transverse, or whether they are even cells (i.e. homeomorphic to balls). By an elaborate scheme of exhaustive and admirable computations MacPherson and McConnell managed to verify [15] the following Proposition 26 (MacPherson-McConnell). Let Σ = {σ} denote Voronoi’s cell structure on S4 . The sets M (σ) := M ∩ σ are closed cells, and any finite union of these cells forms a regular cell complex. Thus through intersections and divine benevolence MacPherson and McConnell identify an equivariant regular locally finite cell structure on M . 10.3 Barycentric retracts Let Γ be a finite index torsion-free subgroup of Sp4 Z (to be precise we require Γ be neat (c.f. [15], pp.619)). Then Γ acts freely on M and (by neatness) acts freely on the boundary components of the compactification M . The important fact required by MacPhersonMcConnell is that the regular locally finite cell structures Σ on M , M described above descend modulo the action of Γ to finite regular cell structures Σ/Γ on the quotients M /Γ , M/Γ . Following [15] we define cells σ in Σ/Γ to be of ‘boundary-type’ if they intersect ∂M = M \ M . Set T = TΓ to be the full subcomplex spanned by the cells in Σ/Γ having ‘boundary-type’. If we consider now the first barycentric subdivision Σ+ , and let W be the full subcomplex of Σ generated by vertices in Σ arising from cells in M − T , then we find W to be a strong deformation retract of M − T (c.f. Lemma 8.7 in [15]). 36 10.3. Barycentric retracts Remark 27. The regular cell structure Σ on M , M do not descend to regular cell structures modulo Sp4 Z. This obstructs us from following [15] and deforming via barycentric retracts. Moreover one can see that no Sp4 Z-equivariant deformation on M can descend to [15]’s barycentric retracts for the simple reason that these retracts move irreducible torsion points (i.e. points of the symmetric space corresponding to lattices whose lattice-isometries are irreducible and nontrivial). 37 Chapter 11 Conclusion The wellrounded retract Xn of Sn is a tantalizing construction. For one, its existence (as a strong spine) cannot be established a´ priori by the present theory, nor could its shape be detected. However the retract itself is geometrically self-evident and compelling. The author throughout the course of his thesis endeavoured to find a comparable retract of hg , and the search continues. The principal merit of this thesis, in the author’s opinion, consists in establishing some basic facts concerning the lagrangian systole function sysL . As outlined in Remark 10, the author hopes this thesis will serve as partial foundation to studying the systolic geometry of the symplectic tori. From this point-of-view, there remain several basic open questions in this regard. However, even within the realm of group cohomology, several important problems remain. Foremost there remains the question of constructing a strong spine for Sp2g Z. Secondly, Stallings theorem 24 deserves a proper resuscitation. The result is rather fundamental, and yet written in an obscure and (respectfully) ‘old-fashioned’ style. There remains the problem of establishing the basic ‘obstruction’ theory which it indicates. Finally with respect to the Schottky problem and riemann surfaces, it would be interesting to determine whether or not sysL has ‘restricted’ behavior on the period locus in the sense of [6]. 38 Bibliography [1] A. Ash. On eutactic forms. Canada. J. Math, 29(5):1040–1054, 1977. [2] A. Ash. Small dimensional classifying spaces for arithmetic subgroups of general linear groups. Duke Math. J., 51(2):459–468, 1984. [3] C. Bavard. Classes minimales de r´eseaux et r´etractions g´eom´etriques ´equivariantes dans les espaces sym´etriques. J. London Math. Society, 64(2):275–286, 2001. [4] A. Borel and J.-P. Serre. Corners and arithmetic groups. Comm. Math. Helv, 48:436– 491, 1973. [5] K. Brown. Cohomology of groups. Graduate Texts in Mathematics 87. Springer-Verlag, 1982. [6] P. Buser and P. Sarnak. On the period matrix of a riemann surface of large genus. Invent. Math, 117:27–56, 1994. [7] J. W. S. Cassels. An introduction to the geometry of numbers. Die grundlehren der mathematik, Band 99. Springer-Verlag, 1959. [8] F. Connelly and T. Ko´zniewski. Finiteness properties of classifying spaces for proper γ-actions. Journal of Pure and Applied Algebra, 41:17–36, 1986. [9] J. H. Conway and N. J. A. Sloane. Sphere packings, lattices and groups. Grundlehren der mathematischen Wissenschaften 290. Springer-Verlag, 1998. [10] D. R. Grayson. Reduction theory using semistability. Comm. Math. Helv., 59:600–634, 1984. [11] D. Grenier. Fundamental domains for the general linear group. Pacific. J. Math, 132(2):293–317, 1988. 39 Bibliography [12] P. Gunnells. The Topology of Hecke Correspondances. PhD thesis, Stanford University, May, 1994. [13] J.Martinet. Perfect Lattices in Euclidean space. Grundlehren der mathematischen Wissenschaften 327. Springer-Verlag, 2003. [14] J. J¨ ost. Nonpositive curvature: geometric and analytic aspects. Lecture notes in mathematics, ETH Zurich. Birkhauser, 1997. [15] R. Macpherson and M.MacConnell. Explicit reduction theory of siegel modular threefolds. Invent. Math, 111:575–625, 1993. [16] A. Pettet and J. Souto. Minimality of the well-rounded retract. Geometry and topology, 12, 2008. [17] C. P. Rourke and B. J. Anderson. Introduction to Piecewise-linear topology. Ergebnisse der mathematik, Band 69. Springer-Verlag, 1972. [18] W. Schmidt. On heights of algebraic subspaces and diophantine approximations. Ann. Math, 85(3):430–472, 1967. [19] C. L. Siegel. Topics in complex function theory, vol III. Tracts in mathematics no. 25. Wiley-Interscience, third edition, 1979. [20] C. Soul´e. The cohomology of SL3 Z. Topology, 17:1–22, 1978. [21] J. Stallings. The embedding of homotopy types into manifolds. preprint, available from J.Stallings homepage, 1985. 40
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Geometric retracts of Siegel's upper half space Martel, Justin 2013
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Title | Geometric retracts of Siegel's upper half space |
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Martel, Justin |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | The purpose of this thesis is to construct a codimension 1 cocompact Sp₂gℤ-equivariant strong deformation retract Wg of Siegel's upper half space hg . This yields a partial contribution towards the problem of constructing a strong spine for the real linear symplectic group Sp₂gℝ. |
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Language | eng |
Date Available | 2013-04-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0071926 |
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Master of Science - MSc |
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Mathematics |
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Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2013-05 |
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Scholarly Level | Graduate |
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