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Orderable groups and topology Wilmarth, Constance 2000

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ORDERABLE GROUPS AND TOPOLOGY by CONSTANCE WILMARTH B.Sc. (Mathematics) University of Oregon, 1997 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics WevaccepA this thesis as conforming to thfe teiJ*rfrefl standard  THE UNIVERSITY OF BRITISH COLUMBIA November 2000 © Constance Wilmarth, 2000  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University- of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Mathematics The University of British Columbia Vancouver, Canada Date  Z<W  Abstract This thesis examines some connections between topology and group theory, in particular the theory of orderable groups. It investigates in close detail some landmark results on this mathematical interface, beginning with Holder's Theorem, and touches upon some recent results in this expanding field of research. Simply stated, Holder's Theorem asserts that Archimedean orderable groups are none other than subgroups of the group of real numbers under addition. Since Holder proved this in 1902, only one significant refinement, due to Paul Conrad, has been made, so these powerful theorems provide the foundation for our understanding of orderable groups. In particular this understanding has served topologists well. This thesis is mostly a distillation of work done in connection with topological applications of the theory, which are surprisingly varied and diverse. Burns and Hale's work on local indicability and right orderability is considered, as well as Bergman's study of the universal covering group of S L ( 2 , R ) . In addition N . Smythe's extension of a classical result of Alexander's via the left orderability of the fundamental groups of certain surfaces is investigated.  ii  Table of Contents Abstract  ii  Table of Contents  iii  Acknowledgements  iv  Chapter 1.  Introduction  1  Chapter 2. Holder's Theorem  4  Chapter 3.  Some Group Theory and Satan's Parkade  10  Chapter 4.  Knot Theory and Orderable Groups  24  Bibliography  33  iii  Acknowledgements I would like to thank my supervisor Professor Dale Rolfsen for his unfailing patience and assistance throughout this project, and Bert Wiest, who was the second reader for the thesis and made many helpful suggestions. Thanks are also due to Colleen Robles and Boris Tschirschwitz.  iv  Chapter 1 Introduction  In 1994, P. Dehornoy [11] proved that the braid group B is right orderable: the proof, n  considered difficult, conveyed no geometric intuition, and a second proof approaching the result from this latter perspective has since been added to the literature. This startling result, which opened the door to further insight into braid theory (for instance, it has recently been shown that P , the pure braid group, is bi-orderable) was the inspiration and n  genesis of this master's thesis, which undertakes a retrospective of some important results in topology that have involved, specifically, the theory of orderable groups. The diversity of these topological applications suggest some of the power andflexibilitythat order considerations bring to bear on topology, and contains the promise that this conjunction of mathematics will continue to yield interesting research in the future. We begin with the following definition.  Definition 1 A group G is said to be right orderable if there exists a total order relation < on the set G such that a < b implies ay < by for all a,b,y G G; in this case we say that < is invariant underrightmultiplication. A group G is left orderable if it is invariant under left multiplication. A group G is bi-orderable or orderable if a < b implies xay < xby for all a,b, x,y e G.  Now it is easily shown that if G is right orderable, then it can also be left-ordered (not  1  Chapter 1. Introduction necessarily by the same ordering!), and vice versa; but groups abound which possess onesided orderings but which are not bi-orderable. A well-known example is the fundamental group G of the Klein bottle, which has presentation G = (a, b \ aba' = b' ) = Z xi Z. 1  1  It is known (for details, see [9] ) that if N < G is right orderable and G/N  is right  orderable, then G is right orderable. We have, in the example of the Klein bottle, Z = (b) normal in G by virtue of the semi-direct product construction, and G/{b) = Z, as (a, b | aba~ = b' ,6 = 1} = (a; —); so G is right orderable. But if G were orderable we l  1  would have b>e=>ab>a=$- aba  -1  >e  b' > e, a contradiction as b > e => 6 1  _1  < e.  It is natural to inquire, in the wake of such definitions and distinctions, why orderable groups might be worth studying. To give an idea of the value of orderability as a mathematical tool, we note that if G is right orderable, then it is torsion free. For take g ^ e, say g > e. Then g > g => g > g 2  3  2  ... =4> g > e for every n £ N, implying that n  g zfz e Vn. It is known that if G is a right-orderable group and R is a domain, then the n  group ring RG has no zero divisors and only trivial units. It is also known that if G is an orderable group, then ZG embeds in a division algebra. So knowing that a group is right-orderable or orderable gives important information, not just about the group itself, but about some related mathematical structures as well. The purpose of this thesis is to investigate the theory of orderable groups in relation to topology, and is organized as follows.  The second chapter is devoted to Holder's  Theorem, which states that a group G is and Archimedean if and only if it is isomorphic to a subgroup of (R, +).  Paul Conrad's refinement of this result, which allows for a  weakening of the hypotheses of the theorem to right orderability and Archimedean, is noted. The next chapter positions right orderability in the larger group theoretic context with respect to the class of locally indicable groups. In this chapter major contributions by Burns and Hale and Bergman are investigated. In the final chapter, orderable groups are featured in their interactions with knot theory, and Neville Smythe's generalization  2  Chapter 1. Introduction of a result in classical knot theory is studied.  3  Chapter 2 Holder's Theorem  As the study of orderable groups properly begins with Holder's characterization of Archimedean ordered groups, and as the proof of the result is obtainable in English only in the form of a brief sketch [22], it seems appropriate to begin this study of orderable groups and topology by working out the details of Holder's argument. But first we will need some preliminary definitions.  CG  Definition 2 Define the positive cone P  of an orderable group G to be the set  {x G G : x > e } .  Definition 3 Let < be an order on an orderable group G. Forx  G G define the  absolute  value of x as  (2.1) -l  if x < e.  For x,y G G write x <C y (in words, x is infinitely smaller than y) to mean that x < \y\ n  for all n G Z. If neither x <^ y nor  write x ~ y.  Observe that for every x£G,x~x&sx ytx.  Moreover, the relation ~ is symmetric  l  by definition. Suppose finally that x ~ y and y ~ z. Then there exist s , t e Z satisfying 4  Chapter  2.  Holder's  Theorem  X > \y\ and y > \x\, so x s  t  st  = (x Y > s  > y* > \z\; it follows easily that x ~ z. So  ~ is an equivalence relation; the equivalence classes so defined are called Archimedean classes.  Definition 4 If all the non-identity elements of G are equivalent, then (G, <) is an Archimedean group, that is, an Archimedean order is one in which G has only two Archimedean classes.  Note: in an Archimedean group, for all x, y such that x < y, x ^ e there exists n £ Z with \x \ > y. n  Now it is not the case that every ordered group is Archimedean. For instance, consider (Z x Z, +) with the dictionary order (ai, bi) < (a , b ) if a\ < a , or if ai = a and b\ <b . 2  Observe that (0,x) «  2  2  2  2  (y,z) if y > 0.  Theorem 1 (Holder) Let < be an ordering on an orderable group G. Then < is an Archimedean order if and only if G is order-isomorphic to a subgroup of the additive group of the real numbers under the natural order.  Proof.  J  (<=) (K, <) is Archimedean, and so any subgroup of it is. (=>) The strategy is to show first that G is an abelian group, and then to use this fact to define an isomorphism. Assume that < is an archimedean order on G. The claim is that G is abelian. For consider any t G G such that t > e.  5  Chapter 2. Holder's Theorem  Case 1. Suppose that e < t < x for every x G P C G. Then G Archimedean implies the existence of an integer n for each x such that x  t* < x < t *  => e < xt~  •  e = xt~ ,  n  n  +l  nx  <t  nx  since t is smaller than any element in the positive cone,  =>• x = t  Ux  =>G=(t)  So G is cyclic and hence abelian. Case 2. Assume now that there exists u G G such that e < u < t. If t < u , that is, if 2  e < u < t < u , then 2  t < u =>•  <u  ,  l  2  u~Hu~ < e  => (wH) < t. 2  So relabelling we have found u (choose u~ t above) such that e < u < u < t; thus in l  2  all cases we may assume that there exists u such that e < u < u < t. Now if G is 2  not abelian, then there exist x,y G G such that commutator [x, y] ^ e. Without loss of generality take [x, y] > e. Then with [x, y] playing the role of t in the inequalities above we obtain, as G Archimedean implies that there exist m , n G Z such that u < x < u m  and u < y < u , that [x,y] < u~ u~ u u n  n+1  m  n  m+l  n+l  = u , which is a contradiction as 2  e < u < u < [x, y]. Therefore we conclude that G is abelian. 2  6  m+l  Chapter 2. Holder's Theorem  The next step is to show that G is order-isomorphic to a subgroup of (R, +) by defining explicitly an isomorphism G -» (R, +) taking x G G to the limit of a certain sequence to be defined for each x. First fix some t > e, t G G. Given x G G and n G Z , there exists +  m G Z (m dependent on x,n,t) such that t  < x" < t  m  define ( (x) = | e S so that t  < x" <  m  Claim: l i m ^ o o ( (x) n  . So for each x, for every n,  m+l  t .  2  n  2  m+1  exists (for each x).  To prove the claim, it suffices to show that {( } is a Cauchy sequence, that is, it suffices n  to show that 1 Cn < Cn+1 < Cn + 7^ •  Lemma: For any four elements a, b, c, d of a bi-orderable group with a < b and c < d, we have ac < 6d (and so, in particular, a < b for every n e R n  n  Proof of Lemma: ac < be < bd. Now, if t  m  <x  2n  <t \  then it follows from the lemma that t  m+  2m  <x  2n+1  < t^ 2  .  m+l)  This implies that 2(m+1) _ m + 1  _m_2m C"  —  2 "  —  2 n T T  -  <  2^+1  _  2 "  J_ ~  +  2"'  as desired. Thus the sequence is Cauchy, hence convergent. We are therefore in a position to define a map 4>>•: G —>• (R, +), </>(a;) = lim^^oo Claim: (j) is order-preserving (that is, x < y =>• </»(x) < 0(y)). Proof of claim.  7  Cn(^)-  Chapter 2. Holder's Theorem Let x,y G G,x < y. Now we already saw that ( (x)&nd Cn(y) are mbnotonic increasing n  sequences. Thus to show that <f)(x) < (f)(y), it suffices to show that ( (x) < Cn(y) f °  r  n  every n. But for every n we have x < y  rc " < y " by the lemma. If m - and m 2  x  are the largest possible integers satisfying t  <x  mx  2 <- y2 implies that m <m . n  x  and t  2n  my  y  < y , then the fact that 2n  It follows that  n  x  2  y  C»(aO =  ^ < ^  = Cn(y).  Thus <p is order-preserving. Claim: ^ is a homomorphism. Proof of claim. We we must show that 4>(xy) = <f)(x) + (p(y). As usual, there exist m , m , ntX  n y  for every  n G N such that +1  -f.m ,y < y2  < £rrc ,j, + l  n  n  n  These, together with the lemma and the fact that G is abelian, imply that j.m ,n+my,n <  y™ 2  x  g  0  tm,,-.+m»,»  <  (2^)2" <. m , + m , + 2 i  as  B  s  n  Cn(x) + Cn(y) and ^ ™«>"+ = ( ( ) m  +  2  2  n  x  therefore ( (xy) lies between *-"+ v-" m  j  a  n  d  n  + (() n  y  + 2^ \ n  m  2"  Then taking n  oo we have  Chapter 2. Holder's Theorem 4>(xy) = <f>{x) + 4>{y). So <?!) is a homomorphism. Claim: <j) is injective. Proof of Claim. We show ker<b = e as follows. Assume x G P . Case 1: t < x. There exists m such that t  < x* <t  mn  2  n  mn+l  m + 1 > 2" Vn (as t < x). It follows that m > 2" n  n  lim _ n  >00  for every n, implying that  l=>ff>l-^>0Vn>l=^  C (x) 7^ 0 as ( (x) is monotonically increasing. Thus (j>{x) ^0. n  n  Case 2: x < t. Here there exists N G N such that t < x ". Again m satisfies t 2  <  mn  n  x  <t"  2n  2-  m  n  N  Vn  +1  G  N, implying that  f, » <  =>m + l > 2 2~ =>m > 2 2~ n  N  n  n  Finally, x £ P, x ^ e  N  n  a;  - 1  G P  ^ -  N+in N)  m  x  <  t™n+i_  - 1 ^ ^ > ^ - J  r  B  u  ^  t  t  ^  m  n  +  x  >  l i m ^ Cn(x) > 0.  = >  </>(z ) = -<i>{x) ^ 0 (as x' -1  >  1  ker<j> by the above)  =*> ^(x) ^ 0. • This, then, is the proof of Holder's Theorem. Paul Conrad [9] added a remarkable footnote in 1959:  Theorem 2 (Conrad) / / G is right orderable and archimedean, then G is orderable. Thus by Holder's Theorem, G is order-isomorphic to a subgroup o/(R,+).  Conrad's proof is short and simple, and depends upon the fact, easily verified, that G is orderable if and only if P is normal in G.  9  Chapter 3  Some Group Theory and Satan's Parkade  In attempting to understand right orderable groups it becomes important to situate them in relation to other classes of groups. Within this context the class of locally indicable groups assumes particular prominence, as it has been shown that locally indicable implies right orderable, but right orderable does not imply locally indicable. We now turn to these results with a view to understanding a remarkable topological application, the universal covering group of SL(2,R), which we will denote by SL.  Definition 5 A group G is said to be locally indicable if each of its non-trivial finitely generated subgroups can be mapped homomorphically onto a non-trivial subgroup o / Z . More generally, if X is a class of groups closed under forming isomorphic images then G is locally X-indicable if every non-trivialfinitelygenerated subgroup admits a nontrivial homomorphism onto a group in X.  The following results are stated without proof ([9], [8]):  Theorem 3 (Conrad) A group G is right orderable if and only if for everyfinitesubset {xi, ...,x } C G that does not contain e, there exist Ci = ± 1 , 1 < i < n, such that e does n  not belong to the subsemigroup of G that is generated by { x f , ...,x ™}. e  10  Chapter 3. Some Group Theory and Satan's Parka.de T h e o r e m 4 ( B u r n s a n d H a l e ) If G is locally RO-indicable, then it is right orderable.  C o r o l l a r y : If G is locally indicable, then G is right orderable, as Z is right orderable. A n example of a right orderable group that is not locally indicable will be found in the universal covering group of SL(2, E). But this is not a simple construct and will require some additional work to understand. We begin with some definitions.  D e f i n i t i o n 6 A t o p o l o g i c a l g r o u p G is a group that is also a topological space, satisfying the requirements that the multiplication map m : G x G —> G sending (x, y) to x • y and the inversion map sending x to x~ are continuous. l  A n important example: the set M„(E) of n by n matrices is a euclidean space of dimension n . As the determinant function det : M„(E) —> E is continuous, dei (0) is a closed _1  2  set, and its complement, the group G L ( n , E), is thus an open subset of E " . Note that 2  matrix multiplication, which is given by polynomials in the coefficients, is continuous, and the inversion map, which by Cramer's rule is a rational function of the coefficients, is also continuous.  D e f i n i t i o n '7 Let G be a topological group with operation •, and assume thatp : G —> G is a simply connected covering space of G.  Then if G is locally path-connected, there  exists a multiplication map rh on G relative to which G is a topological group and p is a  homomorphism. The group G is called the universal covering group of G.  Let V : G x G ->• G x G be the map p x p, and let e e p (e) be selected, where _1  e = l o For the existence of rh required by the definition, we note that by the lifting theorem mV lifts to rh with rh(e,e) = e if and only if (mV)*(iTi(G m*V*{iTi(G x G, (e, e))) C p*(iri(G,  e)).  x G, (e, e))) =  Since G x G is simply connected the containment 11  Chapter 3. Some Group Theory and Satan's Parkade is automatic and we are guaranteed the lift m, which as the lift of a continuous map is continuous. For the condition that p is a homomorphism,. the multiplication map fn is defined as follows. Let x,y to y.  G G.  Choose paths a from e to x and ft from e  Let a(t) = pa(t), and (3(t) = pJ3(t). Then take the path j(t) = a(t) • /3(t)  (where • is the group operation in G, not a path product) and lift it to a path 7 in G that begins at e. Then x • y is defined to be the endpoint of 7 . Note that we have p(x • y) = p(7(l)) = 7(1) = a ( l ) • 0(1) = P&(1) • P(1) = p{x)p(y). P  Now SL(2,K) is not its own universal covering group, as we establish in the following claim. Claim: 7r (SL(2, K), I) = Z (where J is the identity matrix). 1  Proof of Claim. Recall the following definitions.  D e f i n i t i o n 8 An o r t h o g o n a l m a t r i x is a matrix A for which AA = I. The set O(n) 1  of orthogonal (real) matrices forms a subgroup of G L ( n , R )  ;  and SO(n) = {A G O(n) |  det(A) = 1}, the special orthogonal group, is a subgroup o / S L ( n , R).  It is a standard result that O(n) is compact.  D e f i n i t i o n 9 A subspace Y of a space X is called a d e f o r m a t i o n retract if there exists a continuous retract r : X —>• Y such that the identity map on X is homotopic to the map i o r, where i is the inclusion of Y in X.  A subspace Y C X is a strong  d e f o r m a t i o n retract of X if there exists a continuous map H : X x I —> X such that H{x, 0) = x for every x e X, H(x, 1) e Y Va; G X, and H(s, t) = s Vs G Y and V i G I.  12  Chapter 3. Some Group Theory and Satan's Parkade Recall that if Y is a strong deformation retract of X, with y 6 Y, then the inclusion 0  map i : (Y, y ) —> (X, y ) induces an isomorphism of fundamental groups. W i t h this 0  0  artillery we are ready to prove that S L ( 2 , E ) is not simply connected, by showing first that TTI(SL(2,R)) = TTI(S0.(2)), and then that TTI(S0(2)) = TTI(5 ). 1  Subclaim #1: SO(2)  S L ( 2 , E ) is a strong deformation retract.  Proof of Subclaim. We construct the homotopy as follows. Let A £ S L ( 2 , E ) , let e = (J),e = (°). Let x  2  Vi = Aei,V2 = Ae . Since det(A) = 1, A is invertible, v\ and v are linearly independent. 2  2  We use the Gram Schmidt process on v and v to orthnormalize via three homotopies: x  1). Set hi(t) = tfy  2  + (1 -t)vi, sending v —>  = v'{.  x  2. Replace v by v' = v — ^ p j p W i via the map h (t) = v — t(v • v")vi. 2  2  2  2  3. Replace v' by v' ' = ^ 2  via h {t) = t ^  2  3  2  2  + (l- t)v' . 2  After reparametrizing these three homotopies we obtain the single homotopy desired. Thus to prove that 7Ti(SL(2,E)) = Z , it suffices to show that S0(2) is homeomorphic to S. 1  Subclaim #2: SO(2) ^ 5 . 1  Proof of Subclaim. We will use the fact that for X compact and Y Hausdorff, if / : X —> Y is continuous, one-to-one, and onto, then / is a homeomorphism. Since det: 0(2) —> E is continuous, d e £ ( l ) = S0(2) is closed, hence compact. Moreover S -1  is Hausdorff. We define / :  1  S O ( 2 ) —> S , f(A) = A(l). 1  Then / takes a matrix to where it moves the point e , and ±  since SO(2) C 0(2) preserves lengths of vectors / maps S  l  13  into itself. Note that / is  Chapter 3. Some Group Theory and Satan's Parkade clearly onto, as ^ cos 9 — sin 9 ^ sin 9 G S0(2) for every 9 G [0, 27r). A(l)  cos 6  Moreover the invertibility of elements in SO(2) gives  = B(l) <!=> 1(1) = A^BQ)  B = A. Continuity is also immediate from the  continuity of the sine and cosine functions and the fact that elements of SO(2) acts as rotations on S : for if Rg G SO(2) is rotation by angle 9 with image Rgei * = (cos 9, sin 6), 1  we can always choose rotation by angle (j), R<p, with 9 < 4> < 9 + e to obtain images that are arbitrarily close. Summarizing: we have seen that SL(2,R) is not simply connected; and in the ensuing discussion we will investigate more closely the geometry associated with its covering group.  Definition 10 A (right) group action is a map X x G—> X, (x,g) H-> x-g satisfying x • e = x and x • (hg) = (x • h) • g for all g,h G G,x G X. The kernel of the action is the set {g G G | x • g = x Vx G X}. An action is faithful provided that the kernel is trivial. In equivalent terminology that is often used: an action is effective if (x • g = x Vx G X)  g = e.  Of course, a left action is defined in the obvious way. If a set X admits a right (respectively, left) group action, X is said to be a right (respectively, left) C7-space. It is easy to check that if X is a left C7-space and if we define x • g = (# ) • x for all x G X and -1  g G G, then X is a right G-space. To show that G is right orderable, we will need the following characterization of right orderable groups: 14  Chapter 3. Some Group Theory and Satan's Parkade Proposition: G is right orderable if and only if G acts effectively on an ordered set by order-preserving bijections. Proof: (=>•) Let G act on itself by right multiplication. Then right invariance gives precisely the order-preserving property desired. (<=) Assume now that G acts effectively on an ordered set X by order-preserving bijections. Assuming the Well Ordering Principle, let X be well-ordered by some order (X,-<).  The idea is to define an order (G,<) in the following way. Since G  acts effectively on X, if g, h e G are such that g ^ h, then there exists x E X such that x-g  ^ x • h. Under the well-ordering, there exists a minimal such x at  which g and h differ: denote this  Say g < h provided  For right invariance assume that g < h, that is,  • g < {g,h) • h. x  • g < {g,h) • h; then for any x  / € G we require gf < hf, but in fact X( f,hf)'' 9f < (gf,hf) ' hf follows at once x  g  from the assumption that G acts in an order-preserving manner. Finally we show that g < h,h < k =>• g < k. There are a couple of cases to consider. If x^h) = {h,k), then x  easily X' ) • g < X( 9th  9th)  that X( h) ^  £(/i,/c);  9t  • h = X' ,k) • h < x^ ) • k = x^ h  let us first suppose that  but h and k must agree at x^^ {g,h) • 9 < ( ,h)• h =  x  h%k  x  g  -< x^ ),  • k, so g < k. Otherwise assume  h)  •< X( , )- Now g and h differ at h k  so necessarily g and k disagree there, that is,  k  • k. At no point less than x^ ^) do g and k differ, since k = h g  and h = g prior to X( h)- Thus we obtain g < k. If on the other hand x^ ) 9t  k  g = hat X( ) and therefore x> ) • g = x^ htk  hjk  k)  • h < x^  at which g, h, and k differ, we again have g < k. •  k)  • k. Since  < { ,h), then x  g  is the first point  1  Now it is evident that SL(2, R) acts by matrix multiplication on the set of rays through 0 in the plane: if A e SL(2, R), and x = (x, y) is a point on a ray p through the origin, A(tx) = tA(x) for every t e R by linearity, so rays are indeed taken to rays. The  15  Chapter 3. Some Group Theory and Satan's Parkade  kernel of this action is trivial, since if Cl denotes the set of rays through the origin and A G SL(2,R) fixes every ray through the origin, then certainly AQ) = (*•) for some A > 0, and  = (°), some (5 > 0; but then det(,4) = 1  (3 = A " . Now A fixes every 1  ray, so the point x = (}) is mapped to (^_i) = t(l) for some t G R =>- A = 1 and A = I. +  In the action just described, the angle through which a ray is moved is specified only modulo 27T for A G SL(2,'R), but an element of the covering group S L which projects down into SL(2,]R) may be thought of as a linear transformation of the plane in which the angle a ray is moved is specified continuously in the real number s: if A moves p by angle s, there exist elements of S L that move p by angles s 4- 27r/c for every k G Z. Thus the action of S L on Cl is certainly not faithful (there are infinitely many elements that project to / G SL(2,'R)), but if we "unwind" the.circle of rays through the origin and identify them with R, then S L indeed acts faithfully on this set. If one is mathematically inclined to lurid geometrical visions, one can visualize this set-which is the set of rays through 0 in the infinite-sheeted branched covering of the plane with branchpoint 0-as a sort of infinite parkade spiralling above the plane (in which parking space 2nk is directly overhead parking space 2ir(k — 1)). Hence the name Satan's Parkade. The claim is that S L is right orderable, preserving the usual ordering of the line R. We will use the proposition above, showing that S L acts effectively on R by order-preserving bijections. Note that SL(2, R), in acting on the rays through the origin of the plane, can also be considered as acting on 5 by orientation-preserving homeomorphisms, where 1  x G S is taken by A G SL(2,R) to pffy- So we will prove the more general result that if 1  A : R —>• R is a lift of any homeomorphism A : S —> S , then A is also a homeomorphism; 1  1  and that if A preserves orientation on the circle, then A preserves the ordering on the line.  16  Chapter 3. Some Group Theory and Satan's Parkade For given A a homeomorphism from S to itself, A will be a lift of the map Ap and is l  thus continuous:  R,x~  R,y  0  S ,x  0  1  >-S ,y 1  0  A  0  To see that A has a (continuous) inverse, let B = A~ . That is, B is the lift of the l  map A p, and we obtain pBA = A~ Ap = Ip. Since Ipx~o — Ix = x , pBAfo — x ; _1  x  0  but also plx  0  0  — px = XQ, so BA = I as maps. Analogous reasoning yields AB — I,  0  0  and we can conclude that the inverse of A i s . A ; note that A' - 1  is continuous. Thus  1  homeomorphisms of S lift to homeomorphisms of R. 1  We now assume that A : S —> S preserves orientation, and show that A lifts to an 1  1  order-preserving homeomorphism on R. Consider the points (*) and A^) = ( °^ ) (we c  e  9  will write (cos9,sin9) for convenience). Let Rg be rotation on S by angle —9. Note 1  that Re is an orientation-preserving homeomorphism and that (Re o A ) Q = (J). Let g : S —>• S be this map i?e ° A. Then it suffices to show that g : R —> R preserves order, l  1  where (R,p) is a covering map (that is, pg = gp; we take p(£) = (cost, sint)). We already know that homeomorphisms of the circle lift to homeomorphisms of the line; so as g is either an increasing or a decreasing function it suffices to show that g maps R to R . +  +  Since p is a covering map there exists an open neighbourhood U C S of (J) such that 1  each component of p~ (U) is mapped topologically onto U. So there exists an interval l  V = (—e, e) C R such that p restricted to V is a homeomorphism. Since g is continuous there exists 5i > 0 (take Si < e) such that d(0,y) < 5 =4> d(g(0),g(y)) < e. Then 1  putting Ui = p(—8i,5i) we have Ui C J7. Moreover since ^ is continuous we can choose a neighbourhood U C C/ small enough that 2  x  flf(f/ ) 2  C C/x. Let V - p~ (U ) = (-5,5) for l  2  some 5 > 0. To show that g takes R to itself we need only show that <?(f) > 0. Recall +  17  Chapter 3. Some Group Theory and Satan's Parkade  that g fixes (J) on S , so that on V we have g(0) = 0. To see why #(§) > 0, consider 1  P<Kf)  =  9P(%)  =  9( °s c  f, sin | ) . Since smt is an increasing function at 0 and g preserves  orientation, g(cos §, sin |) will have positive angle measure. It follows that #(§) > 0 on V C R.  Thus orientation-preserving homeomorphisms on S~ lift to order-preserving homeomorl  phisms on R; so finally we can deduce that SL is a a right-orderable group. We obtain additional information about SL in the following Lemma: For any A G SL(2,R), with t and t G R and t < t , if |t - t \ < 2n, then 0  f  0  f  0  f  \A(t ) - A(t )\ < 2ir for every A G SL projecting to A. 0  f  Proof of Lemma: Suppose for a contradiction that \A(tf) — A(to)\ > 27r; let's first assume that A(t ) < A(tf). 0  Then since A is continuous, there exists ti G (to,i/)  s u c n  that  A(ti) = A (t ) + 27r. Letting p € ^ denote the ray of angle t , p/ the ray of angle tf, and 0  0  0  so on, it follows that A(pi) = A(p ) + 27r = A(p ). So we have t < h but A(t ) = A(ti), 0  0  0  0  contradicting that det(A) = 1. (If we assume A (to). > -4(t/), then there exists ti € (to,t/) with A(ti) = A(tf) + 27T and we have ti < t/ but A(ti) = -A(t/), contradiction.) • Now suppose A takes the point t G R to r . Then from the lemma it follows that 0  0  |A(t +27r) — .A(to) | < 27r, and since A is.order-preserving, as well as a bijection, A(to+2ir) 0  must be A(t ) + 2TT = r + 27r. In fact this observation generalizes to the following 0  0  proposition, of which it is the base case: Proposition: If A : R —> R is an order-preserving homeomorphism such that A(Z) = Z, then if A(0) - 0, we also have A(n) = (n) Vn € Z. Proof: Since A(—n) = — A(n) for every n, it suffices to show that the Proposition is true for n € N . Assume that the assertion is true for all k < n — 1, but suppose that A(n) = m,m > n. Then since A is onto, there exists s G N, s > n, such that A(s) = n, 18  Chapter 3. Some Group Theory and Satan's Parkade contradicting that A is order-preserving.  •  Thus we now know that S L acts on 27r-multiples of vectors in fl in predictable fashion, rather than shuffling them around. Our goal is to define a nontrivial subgroup of the covering group of SL(2,R) that will map homomorphically to the identity, thus establishing that our group is not locally indicable. To do this we begin by specifying some matrices of interest in S L ( 2 , R ) . Let  T  a where r € R, s G R . +  1  -r  0  1  \  ,r=i  In fact a  r  1  0  U=  0  is a horizontal shear, while b is a vertical shear. r  One easily computes that a has repeated eigenvalue 1 and eigenspace {(*) r  | t G R},  while b also has repeated eigenvalue 1 though with eigenspace {(°) | t G R}, and c has r  s  eigenvalues i/s and  both positive by the condition on s with eigenspaces {(*) : t G R}  and {(°) : s G R}. One also obtains the following identities:  T  Tl  aa  19  Chapter 3. Some Group Theory and Satan's Parkade  The following easy calculations are also very much to our purpose:  =. I  c a c-i r  a  V ^ 0 W l - r \ / 4 - 0 \ 0  ^  / \ 0  1 / \  = I ^  c: b c r  s  0  V~*  I\  f  0 \  v's v  J\ 0  1  1= 1 I \ 0  0  W  , 4= 0 U l 0  l  1  ^  -sr  I = a sr  1  [ 1 0 ,  =  1=6  )  \ rs 1  Finally, letting  d = aba  0  -1  1  0  we note that  c:  1  -  d~ a d= 1  r  .  |  .  1  0  0 -1  0  /  V  0  Ts  )  \  1  /  0  V  -1 \ / 1 - r \ / 0 -1 \ 0 / \ 0  1 / \ 1  0 /  1 \ / 1 0 \ / 0 -1 \  - 1 0 / \ r l / \ l  0 /  =  0  / 1 0 , |  \ r 1  1 -r  =b  r  .  \ 0 • 1  Now a,b,c,d e SL(2,R) satisfy these relations, but the immediate goal is to show that certain elements in their fibres satisfy the same relations in S L ; for once this is established it will not be too difficult to define a finitely generated subgroup of S L which maps  20  Chapter 3. Some Group Theory and Satan's Parkade homomorphically to the identity in Z, thus showing that S L is not locally indicable. Suppose h G SL(2,R) has a positive eigenvalue A > 0 (relevant aside: a,b,c above all do). Then there exists v G R with hv — Xv, so in terms of the action of SL(2, R) on Cl, h 2  fixes a ray v, and in terms of the action of S L on Satan's Parkade, each element of p~ (h) l  must move v by 2nk, some k G Z (p here is the projection map in the definition of the universal covering group above). We define A ,B , C to be the liftings of of, b , and c r  r  r  s  s  that actually fix the rays in the Parkade (rather than moving them by 27rA;). SO in order for this definition to even make sense we need to check that only one of h's infinitely many liftings to S Lfixesthe rays that map into the eigenspace of A. Moreover, since c  s  has two distinct positive eigenvalues, it is necessary to check that the lift that fixes the rays that map into one eigenspace is the same lift that fixes the rays that map into the other. To see why this latter claim is so, we claim that if /  SL(2,R)  G  has at least one  positive eigenvalue, then / moves every ray through an angle strictly less than it. For let A > 0 be an eigenvalue , with v its eigenvector. Suppose w is a ray moved by angle > it. 1  Note that w ^ —v, as then w is also fixed by / . Thus v and w are linearly independent and form an ordered basis for R , but if / moves w by an angle greater than or equal 2  to it, then / is in fact orientation-reversing, contradicting that det(f) = 1. But since / moves every ray through an angle of magnitude less than it', it cannot be lifted to a map / that fixes a ray v[ of angle v and moves some ray v of angle u by some 2itk, where y  2  2  k > 1. For (assuming without loss of generality that v\ < u ) f is continuous, and by 2  the Intermediate Value Theorem there exists c G R, with associated vector w of angle c with ui < c < v such that f(w) = it, so that / moves w by it, a contradiction. 2  Now to see that only one of h's infinitely many liftings to S Lfixesthe rays that map into the eigenspace of A, suppose that there exist hi,h G p~ (h), and hi and h both fix the l  2  2  eigenvector v of h in the Parkade. We must show that hi = h . Let h — hih^ . Then 1  2  ph = p(hih ) l  2  — p{hi)p(h )-  1  2  = hh~ = / . Consider w G Q,w ^ v and chosen to be l  21  Chapter 3. Some Group Theory and Satan's Parkade linearly independent from v. Then of course ph(w) = w. But then h also fixes w in the Parkade, since we saw above that I cannot lift to a map h that moves v by 0 degrees and w by 2nk, k ^ 0 (else there exists u between v and w that is moved by 7r. We conclude that h is the identity, so hi = h^. Thus it makes sense to speak of the unique liftings A ,B , r  Claim: A ,B , r  r  r  and C as defined above. s  and C satisfy the corresponding identities in SL, namely: s  A  r  A  r,  r+r^  =  A  Q T ,  W  =  gr+r^  =  C A C~  =A ,  C^B'Cs  = B .  r  rs  x  s  rs  Proof of Claim. We already know that in each case, both sides the equation project to the same element in SL(2,R). So by the preceding arguments we simply need to show that both sides of each equation have fixed rays, rather than moving them by some 27rfc. But by definition the right hand side of each equation has fixed rays. Now consider A A . r  rl  Recall that a  r  has eigenspace { s p a n Q } for every r £ R. Since matrix multiplication is composition of transformations the eigenvectors stay fixed. So the same holds true for the chosen lift. This argument applies to B B r  rf  and C C , as well. For the last two equations the left S  S  hand sides are conjugates of elements having fixed rays, hence have fixed rays. If we define D = ABA we obtain in similar fashion the identities:  D~ C D  = c;  l  1  s  22  Chapter 3. Some Group Theory and Satan's Parkade  D- A D  = B  D~ B D  = A.  l  l  r  r  r  r  Theorem: SL is not locally indicable. Proof. Let n e N > 1 and let H = (A,B,C )  < SL. Let $ : H —> (Z,+) be any  n  homomorphism. The identities above give that C AC~  x  n  D~ C D x  n  = A ,C~ BC n  1  = B , and n  n  = C~ . So A,B, and C are conjugate in this group to powers of themselves. l  n  Then ^(CnAC- ) = $ ( A ) 1  n  $(C* ) + $(A) - $ ( C ) = $(A ) n  n  n  = n<f>(A) for  some n > 1, implying that $(A) = 0, with similar arguments holding for the other generators. •  23  Chapter 4  Knot Theory and Orderable Groups  Knot (and braid) theory crop up notoriously in all sorts of different mathematical contexts, so it is hardly surprising that orderable groups should intersect knot theory in nontrivial ways. This chapter undertakes an investigation of a result of N. Smythe which exploits order considerations to generalize a classical theorem of Alexander's pertaining to knots to all surfaces. Once again we begin with some definitions.  D e f i n i t i o n 11 A surface is a connected 2-manifold (that is, a connected Hausdorff  space such that each point P has an open neighbourhood homeomorphic to the open disk D° = {(x, y) G R  2  \x + y 2  2  < I}).  A surface w i t h b o u n d a r y is a Hausdorff space  such that every point P has a neighbourhood homeomorphic either to the open disk or to the half-disk {{x,y) £ D° \ y > 0}. If the boundary is empty and the surface is compact, then it is said to be closed.  D e f i n i t i o n 12 A closed non-self-intersecting polygonal line in R is called a p o l y g o n a l 3  knot. A s m o o t h k n o t is the image of an infinitely differentiable embedding f : S  l  R J(t) = (x(t),y(t),z(t)), with ( f , f , | ) ± (0,0,0). 3  Thus we can envisage a knot as an entwined polygon in R (with finitely many edges), 3  or we can envisage it as an entwined circle in that space. Two polygonal knots K  0  and  Ki are said to be equivalent provided Ki can be obtained from K via a finite sequence 0  24  —r  Chapter 4. Knot Theory and Orderable Groups  of "elementary moves" (or their inverses), where these are as follows: let E and Ej be {  adjacent edges of the polygonal knot K , and assume that the triangle spanned by E 0  {  and Ej does not intersect K in any other points. Then an elementary move is simply 0  the replacement of E and Ej by the third leg of the triangle spanned. On the other t  hand, two smooth knots K and K are equivalent if there exists a one-parameter family 0  x  ft : R —> R ,t £ [0,1] of diffeomorphisms smoothly depending on the parameter t such 3  that  3  fo(K ) 0  = KQ  and  fi(K ) 0  F : R x [0,1] — • R ,F(x,t) 3  3  = K\.  Here, "smoothly depending" means that the map  = f (x) is difTerentiable. The family of diffeomorphisms t  ft is called an isotopy, and equivalent knots are sometimes referred to as isotopic, or ambient isotopic. Note that if K and Ky are equivalent, there is a homeomorphism, 0  namely, / i , between (R ,K ) 3  0  and (R ,Kx).  One might well ask whether the double-  3  barreled definition of knots given above-polygonal versus smooth-reflects two different irreconcilable approaches to knots, or whether the definitions are interchangeable. It is the latter that is true: there is a process called "smoothing" taking equivalence classes of polygonal knots to equivalence classes of smooth knots (for details see [7]). The upshot is that we can choose either definition we prefer to work with.  Definition 13 A knot which bounds a disc in R is trivial. 3  A further technicality that must be broached in any discussion of knots is the issue of projection and knot diagrams, which we treat briefly and informally here. In essence what is required in choosing a "regular" projection for a (smooth) knot is a plane such that: • no more than two distinct points of the knot are projected onto the same point on the plane • the set of such crossing points isfinite,and at any such crossing point the projections of two tangent lines to the knot do not coincide 25  Chapter 4. Knot Theory and Orderable Groups • the tangent lines to the knot project onto lines on the plane, not to points. With polygonal knots some adjustment in vocabulary is necessary: we require that a vertex of the knot is never mapped onto a double point, as well as the conditions above that no more than two points are in the pre-image of any projection point (with at most finitely many such double points). Now there is an algorithm known as "laying down the rope", known at least as far back as Alexander and sometimes attributed to him, demonstrating that for any regular projection p(k) of a polygonal knot k, there is a trivial knot with the same projection (see, for instance, [26]). The procedure is simply to orient p(k), choose a starting point x on p(k), and at each double point P of p(k) with itself, in which arcs a(P) and /3(P) cross, to designate a{P) as an undercrossing if in the chosen orientation a(P) lies between x and arc 0(P). The knot that is obtained will be trivial. The question is whether this result generalizes to all surfaces. One generalization, to 5 x / , is immediate: that 2  is, if k is embedded in S x I and p(k) is its projection onto S x {0}, Alexander's 2  2  Algorithm applies since p(k) is contractible in the complement of a point Q of S . Since 2  S — {Q} = R , the algorithm proceeds exactly as above, and the post-algorithmic p(k) 2  2  will still bound a disc. Here is the theorem in full generality:  Theorem 5 ( N . Smythe) Let S be a surface, orientable or not, compact or not. Let k be a polygonal knot contained in, and contractible in, the interior of S x I, with regular projection p(k) in S x {0}. Then there exists a knot k' C S x [|, |] which has the same regular projection, and which bounds a disc in S x [|, | ] .  What is extraordinary is that the theorem hinges upon the left-invariant orderability of the fundamental group  TTI(S),  SO  that a necessary preliminary involves the (non-trivial)  verification that such orderings exist for every surface other than the projective plane RP (which has fundamental group Z / 2 Z , clearly non-orderable, and will require sep2  26  Chapter 4. Knot Theory and Orderable Groups arate treatment). First we observe that the free group of finite or countable rank has a left-invariant ordering. Various proofs exists: see for instance [12] (another slick approach uses the Magnus map). Note that this result will take care of finite surfaces with boundary (whose canonical polygons.have perforations or "holes" bounded by the boundary curves), since any such is homeomorphic to a disc with strips attached (double strips corresponding to handles, Mobius strips to crosscaps, and single strips for any extra perforations); and such surfaces deformation retract onto bouquets of circles passing through the strips, hence have fundamental groups free of rank equal to the number of strips. In addition this will take care of the case in which the surface is infinite (that is to say, non-compact), since the fundamental group of an infinite surface is free. For the details of this result, see, for instance [32]. The basic idea is to represent an infinite surface T as a nested union ( J T of finite surfaces with boundary; with some care one n  n  inductively establishes that the free generators of iti(J- ) remain free in ir^Tn+i); then, n  since any closed path based at x £ J- is compact, it must lie in some T . n  We have as remaining cases the closed orientable surfaces which have fundamental group Gh = (fli,  b  i  ,  {ai,...,a,k I ai a  b  2  2 2  I  h  Yl ibi i~ bi~ a  •••  a  l  = l),h> 0, and the closed non-orientable Hk =  l  = 1), for k > 1. Note that (ai | a  2  = 1) is the fundamental  group of the projective plane, to be discussed separately, while if h = 0 the group is trivial, corresponding to the surface S , and is thus orderable. Turning to the orientable 2  case first we note that if h > 1 we can define a homomorphism ^ : Gh —* Z given by */j(ai) =  1,^(61) =  0,^ (ai) = ^ (k) = OVi > 1. Note that the kernel K contains h  h  h  bi and aj, bi for all i > 2. In the non-orientable case we define the homomorphism :H  k  Z given by * (ai) = 1, ^(^2) = - 1 , ^(a*) = 0 Vi > 2. Note that if k > 2, fc  the kernel contains all powers of at least one a;. In case k = 2 we observe that the kernel contains {a^a^-  ••a "a *" i  1  2  | £'=1i  s  =  EZiJs}-  27  Chapter 4. Knot Theory and Orderable Groups We are interested in the covering spaces corresponding to these (normal) subgroups of Gh and H . Recall that it is possible to define an action of the group ni{S, X) on the fibre k  set p (a:) for any rc: for chosen x € p (x) and any a € 7Ti(S,x), define x • a G p (a;) _1  _1  _1  to be the terminal point of the unique path class a in S such that p*(&) = a and the initial point of a is the point x. Of course the actions by Gh and H , restricted to their k  respective kernels, are still group actions. The claim is that in each instance p~ (x) l  has infinitely many sheets. To see that this is so, consider first the case k = 2, that is, H = (a, b | a b = 1). Then the kernel of the homomorphism to Z contains such 2  2  k  elements of the fundamental group as a b , a b a b , and (a b Y for any j > 1; for chosen 3  3  3  3  3  3  3  3  x G p (:r) we will denote by Xj = x -(a b y the terminal point of the lift. If Xj = x for _1  3  0  3  0  k  some j, k, where j ^ k, then the respective lifts are identical, so p* {(a b ) ) =p*{(a b y), 3  implying, since  3  3  3  3  is a monomorphism, that (a b Y = (a b ) in H . Since j ^ k this 3  3  3  3  h  2  implies that we have (a b ) 3  group, hence (a b ) 3  3  m  3  = 1 for some m £ Z,  m  But H is not a commutative 2  ^ (a b ) for any n; so this is a contradiction. If k > 2, then there 2  2  n  exists ai, i > 2, and all powers of  in the kernel, and similar reasoning leads us to  conclude that p~ (x) has infinitely many sheets, and if h > 1, the fact that all powers l  of bi are in the kernel gives the same result. In all three cases, therefore, the covering space corresponding to the kernel is an open'infinite surface, so the kernel is a free group, hence left-orderable. Since G/K  is isomorphic to a subgroup of Z and left-orderable,  it follows that H and Gh are left-orderable, since as mentioned earlier in this thesis, if k  K < G, and both K and G/K  have left-invariant orderings, then so does G.  With the left-orderability of the fundamental group of these surfaces now established, we turn to an investigation of the theorem in the case wherein 5 is any surface with the exception of S or RP . 2  2  By assumption, k, a polygonal knot, is contained in and  contractible within M xl, where M denotes the interior of S. Let (M,e) be the universal covering space of M x I. Now it is a standard result of covering space theory that if  28  Chapter 4. Knot Theory and Orderable Groups e : X -» Y is a covering map with X simply connected, then Aut(X) = TVi(Y,yo). We shall also make use of the fact that the universal covering surfaces are the sphere (for the sphere itself, and the projective plane), and the plane, which follows from every nonorientable surface having an orientable surface as a 2-sheeted cover [32]. Thus M = R xl, 2  and  =.7Ti(M X I)  Aut(M)  =  which we know is left orderable. Let x be chosen  ITI(S),  as a basepoint in M and let k\ be the lift of p(k) through x. If we let T denote the u  covering translation corresponding to the element u G TT (M X J), then p(k) is covered 1  by  which we denote by  T (ki), u  k. u  A chosen orientation for  p(k) G  x {0} will induce  M  an orientation in k G R x 0. Let p : R x / —> R x {0} be the projection over p, that 2  2  2  u  is, ep = p. Alexander's algorithm of "laying down the rope" can be generalized as follows. Choose neighbourhoods around the double points of the collection {k } sufficiently small to be u  disjoint and contain only the two simple arcs of the singularity. Then focussing attention on k\, we notice that k might cross itself, or ki might cross some other k , where u ^ 1. x  u  If ki crosses itself at the point P , let a(P) denote whichever of the two arcs precedes with respect to x in the orientation of ki, let /3(P) denote the arc that succeeds. If ki crosses k , v ^ 1, let 7(P) denote the arc at the crossing that belongs to ki, and v(P) v  be the arc belonging to k . Then a simple closed curve k G R x / is constructed in 2  v  x  the following manner. For any point (r, s, 0) G ki, if (r, s, 0) does not belong to any of the singularity neighbourhoods, or if  (r, s,  0)  G OL{P),  or if (r, s, 0)  G  7(P) where P is a  crossing of ki and k such that u > 1, then let (r, s, | ) be a point of ki. If on the other u  hand  (r,  s, 0)  G (3{P),  or if (r-, s, 0)  G  7 ( f ) , where  P  is a crossing of  k  x  and  k  v  such that  v < 1, then let (r,s,\ + t),0 < t < | , be a point of A4, where i = \ at P itself (giving (r, s, | ) as the point above P onfci),and t varies continuously to approach 0 as (r, s, 0) approaches the boundary of the neighbourhood at P . Thus we have fashioned a closed curve ki G R x I such that p(ki) = k . Note that a(P) is sent to an undercrossing on 2  ±  29  Chapter 4. Knot Theory and Orderable Groups k  and ft(P) becomes an overcrossing. Finally we define k = T (ki). The algorithm  u  u  u  is rather dauntingly technical, but roughly speaking (and we return to make this more precise below), we are to think of the curves {k } as suspended in E x I with their 2  w  "altitudes" on the interval dependent on the group element which indexes them: higher up for those whose group element is positive in the group ordering, lower down if the group element is negative. Claim: k and k are disjoint. u  v  Indeed, let Q G k n k . Consider T -i(Q) = P. Since k = T (ki), we have P = u  T -i(Q)  v  C T -iT (ki)  u  u  u  u  G h, but also P C T -iT (ki)  u  u  u  G 'k -i . Thus P G M  v  u  v  Consider also P' = T -i(Q). Since k = T (ki), we have P' = T -i(Q) C T -iT (ki) v  and P' C T -iT (ki) v  v  v  v  v  v  k -i . u  v  G jfei  G k -i , so P' G ki fl k -i . Without loss of .generality, take u > v,  u  v  u  v  u  i.e. u~ v < 1. Then we have p(P) a double point of ki and k -i by definition of p, so l  u  v  by construction P must be in E x | . But turning to consider p(P') a double point of 2  k and  we must have P' G E x | , as 2  x  > 1. But this a contradiction, since  T (P) = T [P') = Q, and covering translations do not affect the third coordinate. Thus U  V  k and k are disjoint for all u,v,u ^ v. u  v  Moreover since by the very definition of k we have u > v implying that any crossing of k u  u  lies above k in E x / , u > v implies that k lies above k ; hence we can isotope the curves 2  v  u  v  {k } such that k C R x (0,,|) if u < 1, h remains in R x [|, | ] , and k CR x (|, 1) 2  w  2  2  u  v  if v > 1; and in all cases p(k ) = k remains a regular projection for all w. Note that w  w  is isolated in R x [|, | ] , and that the construction of ki with specific reference to 2  the double points of ki with itself, coincides exactly with the "laying down the rope" algorithm. Thus by Alexander, k\ is contractible in ( U ^ i k ) , and bounds a nonu  u  singular disc D in R x [|, |] C M, the universal cover. Consider e(ki) C M x I: e(ki) 2  bounds e(D). Since D e l  2  x [|, |] is isolated from the other curves {k }, if e(D) has u  30  Chapter 4. Knot Theory and Orderable Groups  singularities, these will be points of self-intersection. We invoke now a celebrated result: Theorem 6 (Dehn's Lemma) Let M be a PL 3-manifold, compact or not, with boundary which may be empty, and in M let D be a two-cell with self-intersections (singulari-  ties), having as boundary the simple closed polygonal curve C and such that there exists a closed neighbourhood of C in D which is an annulus (that is, no point of C is singular). Then there exists a two-cell D with boundary C semi-linearly embedded in M. 0  By Dehn's Lemma, e(D) may be modified in a neighbourhood of these singularities, and thus within M x [|, | ] , so that e(ki) bounds a non-singular disc. There remains one final surface to attend to. As noted before, the fundamental group of the projective plane is not orderable, so this case requires an extra step that will reduce the argument to that of the Mobius band, which has fundamental group Z (the Mobius band deformation retracts to S ). If p(k) is the regular projection onto S x {0} of a knot 1  in S x I, where S is now the projective plane, one can transform the closed curve p(k) into a finite number of simple closed curves C\, C , C , by choosing open neighbourhoods 2  m  of each crossing point small enough to contain no other crossing point and "splicing" the crossing in the following way: if P is the crossing point of arcs a and /3, and N(P) is the selected small disc neighbourhood, one can choose a point A e a fl N(P) preceding P in the orientation of p(k), and a point B on /3 succeeding P in the orientation. Then one joins A and B by an arc 7 within N(P) that doesn't elsewhere intersect p(k), and by deleting the open arcs AP and PB one obtains disjoint closed curves. If we choose a basepoint x on p(k), p(k) is nomotopic to a product of conjugates of the Ci, namely WiCiW~ ,i < i < m, where W{ is an approach path (the trivial path in l  the case i = 1) from the basepoint x that connects to Ci across the disc neighbourhood and coincides exactly with p(k) elsewhere. It will now be expedient to transcribe this set-up into the language of homology: note that as a closed path p(k) can be considered 31  Chapter 4. Knot Theory and Orderable Groups as a 1-cycle, that is, as a (formal) sum of oriented edges whose boundary is 0. So we have p(k) homotopic to a path product WiCiWi w C w 1  2  is the same in homology as Y1T=\  2  • • • w C w^-,  l 2  m  m  but this latter  since the formal sum of edges is unchanged under  conjugation by the {wi}. Thus since p(k) is contractible, WiCiW^w^w^  • • • w C w^  1  m  m  is contractible, whence YHLI Ci is homologous to 0. It will follow that each d is nullhomologous and thus contractible: for suppose that some Ci is not null-homologous. Since the fundamental group of R P is already abelian, H^RP )  =  2  2  TTI(RP ) = Z / 2 Z 2  implies that an even number of the C, must be non-bounding as Y^HLI Ci ~ 0- Assume that Ci and C are not null-homologous. Recall that R P = N Ug D , where iV is a 2  2  2  Mobius band attached along the boundary of the disc JJ) . If C\ does not bound a disc, 2  it circles around the Mobius band. So C C S — C\ is contained in the interior of D , 2  2  hence is contractible, contradicting the assumption that C is non-bounding. Thus each 2  Ci bounds a disc. Since each d is bounding, each d separates RP into two connected regions, a disc A 2  and the complement of the disc, which contains a Mobius band. Since no Cj intersects any Cj, i ^ j, their corresponding discs will either be disjoint or one will be contained in the other. If Di contains Cj, then Di contains Dj (as Di contains no Mobius band); so we can take Di, D ,D , 2  n  n £ { 1 , m } to be the outermost discs not contained in  any other discs, and set S = (J" Di U N(P), where N(P) is the disc neighbourhood for =1  each crossing P of p(k); S is a proper connected subspace of R P which contains p(k). 2  Since within S we have p(k) homotopic to a product of conjugates of Ci, each of which is contractible in S, it follows that p(k) is contractible in S, hence contractible in the complement of a small disc in S, that is, within a Mobius band contained in S. With this, the argument is returned to the earlier case of the proof, and we are done. •  32  Bibliography [1] J.W. Alexander. Topological invariants of knots and links. Trans. Am. Math. Soc, 1928. [2] George Bergman. Ordering co-products of groups. Journal of Algebra, 1990. [3] George Bergman. Right orderable groups that are not locally indicable. Pacific Journal of Math., 1991. [4] Joan S. Birman. Braids, Links, and Mapping Class Groups. Princeton University Press, 1975. [5] Glen E. Bredon. Topology and Geometry. Springer-Verlag, 1993. [6] Serge Burckel. The well-ordering on positive braids. J. Pure Appl. Algebra, 1997. [7] Gerhard Burde and Heiner Zieschang. Knots. Walter de Gruyter, 1985. [8] R.G. Burns and V.W.D. Hale. A note on group rings of certain torsion-free groups. Canad. Math. Bull, 15 (3):441-445, 1972. [9] Paul Conrad. Right-ordered groups. Michigan Math. J., 6:267-275, 1959. [10] R.H. Crowell and R.H. Fox. Introduction to Knot Theory. Springer Verlag, 1977. [11] Patrick Dehornoy. From large cardinals to braids via distributive algebras. J. Knot Theory and its Ramifications, 1995. [12] C.J. Everett and S. Ulam. Ordered groups. Trans. Am. Math. Soc, 1945. [13] F. Thomas Farrell. Right-orderable deck transformation groups. Rocky Mountain Journal of Mathematics, 1976. [14] N.D. Gilbert and T. Porter. Knots and Surfaces. Oxford University Press, 1994. [15] Marshall Hall. The Theory of Groups. Macmillan, 1959. [16] P.R. Halmos. Naive Set Theory. Van Nostrand Reinhold Co., 1960. [17] O. Holder. Die axiome der quantitat und die lehre von mass. Math.-Phys. Kl, 1901. [18] Vaughan F.R. Jones. Subfactors and Knots. AMS, 1991. [19] V . M . Kopytov and N.Y. Medvedev. Right-Ordered Groups. Plenum, 1966. [20] Peter A. Linnell. Left ordered amenable and locally indicable groups. J. London Math. Soc, 1999. 33  Bibliography  W.S. Massey. Algebraic Topology: An Introduction. Harcourt Brace Jovanovich, 1967. Roberta Botto Mura and Akbar H. Rhemtulla. Orderable Groups. Marcel Dekker, 1977. Kunio Murasugi. Knot Theory and its Applications. Birkhauser, 1996. C D . Papakyriakopoulos. On dehn's lemma and the asphericity of knots. Ann. of Math, 1957. Donald Passman. The Algebraic Structure of Group Rings. John Wiley and Sons, 1977. D. Rolfsen. Knots and Links. Publish or Perish, Inc., 1976. D. Rolfsen and Jun Zhu. Braids, orderings, and zero divisors. Journal of Knot Theory and its Ramifications, 1998. N . Smythe. Trivial knots with arbitrary projection. J. Austr. Math. Soc, 1967. A.B. Sossinsky and V.V. Prasolov. Knots, Links, Braids, and 3-Manifolds. AMS, 1991. E. H. Spanier. Algebraic Topology. McGraw-Hill Book Co., 1966. N . Steenrod. The Topology of Fibre Bundles. Princeton University Press, 1951. John Stillwell. Classical Topology and Combinatorial Group Theory. Springer-Verlag, 1980. V . M . Tararin. On convex subgroups of right ordered groups. Siberian Math. Journal, 1994. V . M . Tararin. On the theory of right orderable groups. Math. Notes, 1994.  34  


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