Group actions of A 5 on contractible 2 -complexes
GGROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES IVÁN SADOFSCHI COSTA
Abstract.
We prove that every action of A on a finite -dimensional contractible complexhas a fixed point. Contents
1. Introduction 12. Fixed point free actions on acyclic -complexes 33. A reduction 64. Brown’s short exact sequence 95. A moduli of representations 126. Quaternions 157. Group actions of A on contractible -complexes 19Appendix A. Equations over groups 20Appendix B. Quaternion valued analytic functions 21Appendix C. Alternative proof of Lemma 5.1 22Appendix D. Alternative proofs using SAGE 23References 251. Introduction
A well-known result of Jean-Pierre Serre [Ser80] states that every action of a finite group on acontractible -complex (i.e. a tree) has a fixed point. By Smith theory, every action of a p -groupon the disk D n has a fixed point. The group A acts simplicially and fixed point freely on thebarycentric subdivision X of the -skeleton of the Poincaré homology sphere which is an acyclic -complex. By considering the join X ∗ A , Edwin E. Floyd and Roger W. Richardson [FR59]proved that A acts simplicially and fixed point freely on a contractible -complex. Moreover,by embedding X ∗ A in R and taking a regular neighbourhood they proved that A actssimplicially and fixed point freely on a triangulation of the disk D . This was the only exampleknown of this kind until Bob Oliver obtained a complete classification of the groups that actfixed point freely on a disk D n [Oli75]. The example of Floyd and Richardson makes clear thatit is not possible to extend Serre’s result to -complexes and is natural to wonder if it holdsfor -complexes. Carles Casacuberta and Warren Dicks [CD92] made the following conjecture(without requiring X to be finite) which was also posed by Michael Aschbacher and Yoav Segevas a question [AS93, Question 3] in the finite case. Mathematics Subject Classification.
Key words and phrases.
Group actions, contractible -complexes, moduli of group representations, mappingdegree.Researcher of CONICET. The author was partially supported by grants PICT-2017-2806, PIP11220170100357CO and UBACyT 20020160100081BA. a r X i v : . [ m a t h . A T ] O c t IVÁN SADOFSCHI COSTA
Conjecture 1.1.
Let G be a finite group. If X is a -dimensional finite contractible G -complexthen X G (cid:54) = ∅ . In [CD92] the conjecture is proved for solvable groups. Independently, Segev [Seg93] studiedthe question of which groups act without fixed points on an acyclic -complex and proved Con-jecture 1.1 for solvable groups and the alternating groups A n for n ≥ . In [Seg94], Segev provedthe conjecture for collapsible -complexes. Using the classification of the finite simple groups,Aschbacher and Segev proved that for many groups any action on a finite -dimensional acycliccomplex has a fixed point [AS93]. Later, Oliver and Segev [OS02] gave a complete classificationof the groups that act without fixed points on a finite acyclic -complex. Before [OS02], A was the only group known to act without fixed points on an acyclic -complex. An excellentexposition on this topic is the one given by Alejandro Adem at the Séminaire Bourbaki [Ade03].In [Cor01], J.M. Corson proved that Conjecture 1.1 holds for diagrammatically reducible com-plexes. The smallest group for which Conjecture 1.1 remained open is the alternating group A .The main result of this paper is the following. Theorem 7.2.
Every action of A ∼ = PSL (2 ) on a finite, contractible -complex has a fixedpoint. From this, using the results of Oliver and Segev [OS02], we deduce the following.
Corollary 7.3.
Let G be one of the groups PSL (2 k ) , PSL (5 k ) for k ≥ or PSL ( q ) for q ≡ ± and q ≡ ± . Then every action of G on a finite contractible -complex has afixed point. Our proof of Theorem 7.2 goes by constructing a nontrivial representation in
SO(3 , R ) of thefundamental group of every fixed point free, -dimensional, finite acyclic A -complex. Thereforewe have the following. Corollary 7.5.
Let X be a fixed point free -dimensional finite and acyclic A -complex and let π = π ( X ) . Then π is infinite or there is an epimorphism π → A . The paper is organized as follows. In Section 3 we prove Theorem 3.6 which says that toprove Theorem 7.2 it is enough to inspect the acyclic complexes of the type considered by Oliverand Segev in [OS02]. The necessary results from [OS02] are recalled in Section 2. In Section 3we also prove Theorem 3.8, which describes a possible path towards settling Conjecture 1.1.In Section 4 we establish the connection between Theorem 7.2 and the following group theo-retic statement, using a result of Kenneth S. Brown [Bro84] in Bass–Serre theory.
Theorem 7.1.
There is no presentation of A of the form (cid:104) a, b, c, d, x , . . . , x k | a , b , c , d , ( ab ) , ( bc ) , ( cd ) , x ax − = d, w , . . . , w k (cid:105) with w , . . . , w k ∈ ker( φ ) , where φ : F ( a, b, c, d, x , . . . , x k ) → A is given by a (cid:55)→ (2 , , , b (cid:55)→ (3 , , , c (cid:55)→ (1 , , , d (cid:55)→ (2 , , and x i (cid:55)→ for each i = 0 , . . . , k . In order to prove Theorem 7.1, in Section 5 we introduce a moduli of representations of thegroup Γ k = (cid:104) a, b, c, d, x , . . . , x k | a , b , c , d , ( ab ) , ( bc ) , ( cd ) , x ax − = d (cid:105) in SO(3) . In Section 6 we view these rotations in S ⊂ H , enabling us to do a degree argumentwhich completes the proof of Theorem 7.1. This proof is inspired by James Howie’s proof of the ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 3 Scott–Wiegold conjecture [How02]. Finally, in Section 7 we put everything together to completethe proof of Theorem 7.2.
Note.
Some of the results presented here appeared originally in the author’s thesis [SC19].
Acknowledgements.
I am grateful to my mentor, Jonathan Barmak, for his constant advice,in particular for suggesting me to work on this problem during my PhD. I would like to thankBob Oliver for his feedback on my thesis. I would also like to thank Kevin Piterman for takinga look at a previous version of this article, and Yago Antolin, Ignacio Darago, Carlos Di Fiore,Fernando Martin and Pedro Tamaroff for valuable conversations.2.
Fixed point free actions on acyclic -complexes In this section we review the results obtained by Bob Oliver and Yoav Segev in their article[OS02] that are needed later.Throughout the paper, by G -complex we mean a G -CW complex. That is, a CW complexwith a continuous G -action that is admissible (i.e. the action permutes the open cells of X , andmaps a cell to itself only via the identity). For more details see [OS02, Appendix A]. We willfrequently assume that the -cells in a G -complex are attached along closed edge paths, this willmake no difference for the questions that we study. A graph is a -dimensional CW complex.By G -graph we always mean a -dimensional G -complex. Definition 2.1 ([OS02]) . A G -complex X is essential if there is no normal subgroup (cid:54) = N (cid:47) G such that for each H ⊆ G , the inclusion X HN → X H induces an isomorphism on integralhomology.The main results of [OS02] are the following two theorems. Theorem 2.2 ([OS02, Theorem A]) . For any finite group G , there is an essential fixed pointfree -dimensional (finite) acyclic G -complex if and only if G is isomorphic to one of the simplegroups PSL (2 k ) for k ≥ , PSL ( q ) for q ≡ ± and q ≥ , or Sz(2 k ) for odd k ≥ .Furthermore, the isotropy subgroups of any such G -complex are all solvable. Theorem 2.3 ([OS02, Theorem B]) . Let G be any finite group, and let X be any -dimensionalacyclic G -complex. Let N be the subgroup generated by all normal subgroups N (cid:48) (cid:47) G such that X N (cid:48) (cid:54) = ∅ . Then X N is acyclic; X is essential if and only if N = 1 ; and the action of G/N on X N is essential. The following fundamental result of Segev [Seg93, Theorem 3.4] will be used frequently, some-times implicitly. We state the version given in [OS02].
Theorem 2.4 ([OS02, Theorem 4.1]) . Let X be any -dimensional acyclic G -complex (notnecessarily finite). Then X G is acyclic or empty, and is acyclic if G is solvable. We denote the set of subgroups of G by S ( G ) . Definition 2.5 ([OS02]) . By a family of subgroups of G we mean any subset F ⊆ S ( G ) whichis closed under conjugation. A nonempty family is said to be separating if it has the followingthree properties: (a) G / ∈ F ; (b) if H (cid:48) ⊆ H and H ∈ F then H (cid:48) ∈ F ; (c) for any H (cid:47) K ⊆ G with K/H solvable, K ∈ F if H ∈ F .For any family F of subgroups of G , a ( G, F ) -complex is a G -complex all of whose isotropysubgroups lie in F . A ( G, F ) -complex is universal (resp. H- universal ) if the fixed point set ofeach H ∈ F is contractible (resp. acyclic). IVÁN SADOFSCHI COSTA If G is not solvable, the separating family of solvable subgroups of G is denoted by SLV . If G is perfect, then the family of proper subgroups of G is denoted by MAX . Lemma 2.6 ([OS02, Lemma 1.2]) . Let X be any -dimensional acyclic G -complex without fixedpoints. Let F be the set of subgroups H ⊆ G such that X H (cid:54) = ∅ . Then F is a separating familyof subgroups of G , and X is an H -universal ( G, F ) -complex. Proposition 2.7 ([OS02, Proposition 6.4]) . Assume that L is one of the simple groups PSL ( q ) or Sz( q ) , where q = p k and p is prime ( p = 2 in the second case). Let G ⊆ Aut( L ) be anysubgroup containing L , and let F be a separating family for G . Then there is a -dimensionalacyclic ( G, F ) -complex if and only if G = L , F = SLV , and q is a power of or q ≡ ± . Lemma 2.8.
Let G be one of the groups in Theorem 2.2 and let X be a fixed point free -dimensional acyclic G -complex. If K ≤ G is not solvable then the action of K on X is fixedpoint free.Proof. Let F = { H ≤ G : X H (cid:54) = ∅ } . Then, by Lemma 2.6, F is a separating family and X is an H-universal ( G, F ) -complex. By Proposition 2.7, we must have F = SLV and therefore X K = ∅ . (cid:3) If X is a poset, then K ( X ) denotes the order complex of X , that is, the simplicial complexwith simplices the finite nonempty totally ordered subsets of X (the complex K ( X ) is also knownas the nerve of X ). Definition 2.9 ([OS02, Definition 2.1]) . For any family F of subgroups of G define i F ( H ) = 1[ N G ( H ) : H ] (1 − χ ( K ( F >H ))) . Recall that if G (cid:121) X , the orbit G · x is said to be of type G/H if the stabilizer G x is conjugateto H in G . In other words, if the action of G on G · x is the same as the action of G on G/H . Lemma 2.10 ([OS02, Lemma 2.3]) . Fix a separating family F , a finite H -universal ( G, F ) -complex X , and a subgroup H ⊆ G . For each n , let c n ( H ) denote the number of orbits of n -cellsof type G/H in X . Then i F ( H ) = (cid:80) n ≥ ( − n c n ( H ) . Proposition 2.11 ([OS02, Tables 2,3,4]) . Let G be one of the simple groups PSL (2 k ) for k ≥ , PSL ( q ) for q ≡ ± and q ≥ , or Sz(2 k ) for odd k ≥ . Then i SLV (1) = 1 . For each family of groups appearing in Theorem 2.2, Oliver and Segev describe an example.In what follows, D m is a dihedral group of order m and C m is a cyclic group of order m . Proposition 2.12 ([OS02, Example 3.4]) . Set G = PSL ( q ) , where q = 2 k and k ≥ . Thenthere is a -dimensional acyclic fixed point free G -complex X , all of whose isotropy subgroupsare solvable. More precisely X can be constructed to have three orbits of vertices with isotropysubgroups isomorphic to B = F q (cid:111) C q − , D q − , and D q +1) ; three orbits of edges with isotropysubgroups isomorphic to C q − , C and C ; and one free orbit of -cells. We have A = PSL (2 ) . The barycentric subdivision of the -skeleton of the Poincaré do-decahedral space is an A -complex of the type given in Proposition 2.12 with fundamental groupthe binary icosahedral group A ∗ ∼ = SL(2 , which has order . The Poincaré dodecahedralspace appears in many other natural ways, for more information see [KS79]. ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 5 Proposition 2.13 ([OS02, Example 3.5]) . Assume that G = PSL ( q ) , where q = p k ≥ and q ≡ ± . Then there is a -dimensional acyclic fixed point free G -complex X , all ofwhose isotropy subgroups are solvable. More precisely, X can be constructed to have four orbitsof vertices with isotropy subgroups isomorphic to B = F q (cid:111) C ( q − / , D q − , D q +1 , and A ; fourorbits of edges with isotropy subgroups isomorphic to C ( q − / , C , C and C ; and one free orbitof -cells. Proposition 2.14 ([OS02, Example 3.7]) . Set q = 2 k +1 for any k ≥ . Then there is a -dimensional acyclic fixed point free Sz( q ) -complex X , all of whose isotropy subgroups are solvable.More precisely, X can be constructed to have four orbits of vertices with isotropy subgroupsisomorphic to M ( q, θ ) , D q − , C q + √ q +1 (cid:111) C , C q −√ q +1 (cid:111) C ; four orbits of edges with isotropysubgroups isomorphic to C q − , C , C and C ; and one free orbit of -cells. We also have A ∼ = PSL (5) , so this group is addressed in both Proposition 2.12 and Propo-sition 2.13. There is no other such exception. Definition 2.15. If G is one of the groups in Theorem 2.2, the Oliver–Segev G -graph Γ OS ( G ) is the -skeleton of any -dimensional fixed point free acyclic G -complex without free orbits of -cells of the type constructed in Propositions 2.12 to 2.14. For this definition, we regard A as PSL (2 ) rather than PSL (5) .Generally, there is more than one possible choice for the G -graph Γ OS ( G ) . Even for G = A ,thought of as PSL (2 ) , the quotient graph Γ OS ( G ) /G is not unique. However in Proposition 3.10we show that Γ OS ( G ) is unique up to G -homotopy equivalence. Moreover, Corollary 3.11 showsthe particular choice of Γ OS ( G ) is irrelevant for our purposes. Definition 2.16 (A construction of Γ OS ( A ) ) . Here we give a construction of Γ OS ( A ) and wefix some notation in regard to this graph. Consider the following subgroups of A : H = (cid:104) (2 , , , (3 , , (cid:105) ∼ = A ,H = (cid:104) (3 , , , (1 , , (cid:105) ∼ = D and H = (cid:104) (1 , , , (2 , , (cid:105) ∼ = D . The graph Γ OS ( A ) has three orbits of vertices whose representatives v , v , v have stabilizers H , H , H respectively. In addition, Γ OS ( A ) has three orbits of edges whose representatives v e −−→ v , v e −−→ v and v e −−→ v have stabilizers H = H ∩ H = (cid:104) (3 , , (cid:105) ∼ = Z ,H = H ∩ H = (cid:104) (2 , , (cid:105) ∼ = Z and H = H ∩ H = (cid:104) (1 , , (cid:105) ∼ = Z respectively.Attaching a free orbit of -cells to Γ OS ( A ) along the orbit of the closed edge path ( e , e , e ) we obtain an acyclic -dimensional fixed point free A -complex of the type given in Proposi-tion 2.12. This complex is, in fact, the barycentric subdivision of the -skeleton of the Poincarédodecahedral space (a simplicial complex having
21 = 5 + 10 + 6 vertices,
80 = 20 + 30 + 30 edges and faces). A concrete isomorphism can be produced by mapping v to the barycentreof a pentagonal -cell ABCDE , v to A and v to the barycentre (midpoint) of AB . For moredetails on this see [OS02, pp. 20-21]. IVÁN SADOFSCHI COSTA A reduction
In this section we rely on the results of Oliver and Segev to prove Theorem 3.6, which allowsus to reduce the proof of Theorem 7.2 to the study of acyclic complexes of the type considered in[OS02]. We also prove Theorem 3.8 which describes a possible path to establish Conjecture 1.1.We first prove some results which will be used to do equivariant modifications to our complexes.
Definition 3.1. If X, Y are G -spaces, a G -homotopy is an equivariant map H : X × I → Y . Wesay that f ( x ) = H ( x, and f ( x ) = H ( x, are G -homotopic and we denote this by f (cid:39) G f .An equivariant map f : X → Y is a G -homotopy equivalence if there is an equivariant map g : Y → X such that f g (cid:39) G Y and gf (cid:39) G X . A G -invariant subspace A of X is a strong G -deformation retract of X if there is a retraction r : X → A such that there is a G -homotopy H : ir (cid:39) X relative to A , where i : A → X is the inclusion. Remark . An equivariant map f : X → Y is a G -homotopy equivalence if and only if f H : X H → Y H is a homotopy equivalence for each subgroup H ≤ G (see [tD87, ChapterII, (2.7) Proposition]). Thus, if f : X → Y is a G -homotopy equivalence, the action G (cid:121) X is fixed point free (resp. essential) if and only if the action G (cid:121) Y is fixed point free (resp.essential).The following lemma allows us to do elementary expansions equivariantly. Lemma 3.3.
Let X be an acyclic -dimensional G -complex. Let H ≤ G and x , x ∈ X (0) ∩ X H .Then there is a G -complex Y ⊃ X , such that X is a strong G -deformation retract of Y and Y is obtained from X by attaching an orbit of -cells of type G/H with endpoints { x , x } and anorbit of -cells of type G/H .Proof.
We attach an orbit of -cells of type G/H to X using the attaching map ϕ : G/H × S → X (0) defined by ( gH, (cid:55)→ g · x , ( gH, − (cid:55)→ g · x . Let e be the -cell of this new orbitcorresponding to the coset H . Since X is acyclic, by Theorem 2.4 X H is also acyclic. Let γ bean edge path in X H starting at x and ending at x . Then we attach an orbit of -cells of type G/H in such a way that the -cell corresponding to the coset H is attached along the closededge path given by e and γ . It is clear that X is a strong G -deformation retract of Y . (cid:3) The following very natural definitions appear in [KLV01, Section 2].
Definition 3.4. A forest is a graph with trivial first homology. If a subcomplex Γ of a CWcomplex X is a forest, there is a CW complex Y obtained from X by shrinking each connectedcomponent of Γ to a point. The quotient map q : X → Y is a homotopy equivalence and we say Y is obtained from X by a forest collapse .If X is a G -complex and Γ ⊂ X is a forest which is G -invariant, the quotient map q is a G -homotopy equivalence and we say the G -complex Y is obtained from X by a G - forest collapse .We say that a G -graph is reduced if it has no edge e such that G · e is a forest. Lemma 3.5.
Let X be a -dimensional acyclic G -complex. If X (1) is a reduced G -graph thenstabilizers of different vertices are not comparable.Proof. Let F = { G x : x ∈ X (0) } and let M = { v ∈ X (0) : G v is maximal in F } . We firstprove, by contradiction, that X (0) = M . Consider v ∈ X (0) − M such that G v is maximal in { G x : x ∈ X (0) − M } . Then since X G v contains v , by Theorem 2.4 it must be acyclic. Since v / ∈ M , there is a vertex w ∈ X G v ∩ M . By connectivity there is an edge e ∈ X G v whose ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 7 endpoints v (cid:48) and w (cid:48) satisfy v (cid:48) / ∈ M and w (cid:48) ∈ M . Since G v (cid:48) ≥ G v and v (cid:48) / ∈ M , by our choice of v we have G v = G v (cid:48) . Since e ∈ X G v we have G v ≤ G e and since v (cid:48) is an endpoint of e we have G e ≤ G v (cid:48) . Thus G e = G v (cid:48) and then the degree of v (cid:48) in the graph G · e (which has vertex set G · w (cid:48) (cid:96) G · v (cid:48) ) is . Thus G · e is a forest, contradiction. Therefore we must have M = X (0) . Toconclude we have to prove that different vertices u, v ∈ M have different stabilizers. Suppose G u = G v to get a contradiction. Since u, v are vertices of X G u which is connected, there is anedge e ∈ X G u and by maximality we must have G e = G u . If u (cid:48) , v (cid:48) are the endpoints of e , we have G u (cid:48) = G v (cid:48) . We have two cases and in any case we obtain a contradiction. If G · u (cid:48) (cid:54) = G · v (cid:48) then G · e is a forest consisting of | G/G e | disjoint edges, contradiction. Otherwise, there is a nontrivialelement g ∈ G such that g · u (cid:48) = v (cid:48) and we have G u (cid:48) = G v (cid:48) = gG u (cid:48) g − . Thus g ∈ N G ( G u (cid:48) ) .Consider the action of (cid:104) g (cid:105) on X G u (cid:48) , which is acyclic and thus has a fixed point by the Lefschetzfixed point theorem. But this cannot happen, since this would imply that (cid:104) G u (cid:48) , g (cid:105) (cid:13) G u (cid:48) fixes apoint of X , which is a contradiction since u (cid:48) ∈ M . (cid:3) Now we prove the main results of the section.
Theorem 3.6.
Let G be one of the groups in Theorem 2.2. Let X be a fixed point free -dimensional finite acyclic G -complex. Then there is a fixed point free -dimensional finite acyclic G -complex X (cid:48) obtained from the G -graph Γ OS ( G ) by attaching k ≥ free orbits of -cells and k + 1 free orbits of -cells and an epimorphism π ( X ) → π ( X (cid:48) ) .Proof. Let F = { H ≤ G : X H (cid:54) = ∅ } . Then, by Lemma 2.6, F is a separating family and X isan H-universal ( G, F ) -complex. By Proposition 2.7, we must have F = SLV . By doing enough G -forest collapses we can assume that X (1) is a reduced G -graph. The stabilizers of the verticesof Γ OS ( G ) are precisely the maximal solvable subgroups of G . Therefore, since every solvablesubgroup of G fixes a point of X , by Lemma 3.5, we may identify X (0) = Γ OS ( G ) (0) . ApplyingLemma 3.3 enough times to modify X , we may further assume Γ OS ( G ) is a subcomplex of X .Finally we will modify X so that for every subgroup (cid:54) = H ≤ G , we have X H = Γ OS ( G ) H .We do this by reverse induction on | H | . Assume that we have X such that it holds for ev-ery subgroup K with H (cid:12) K ≤ G . If H is not solvable, we have X H = Γ OS ( G ) H = ∅ sowe are done. If H is solvable, since Γ OS ( G ) H is a tree (it is acyclic and -dimensional) and X H is acyclic by Theorem 2.4, the inclusion Γ OS ( G ) H (cid:44) → X H is an N G ( H ) -equivariant homol-ogy equivalence. Now since Γ OS ( G ) H is a tree we can define an N G ( H ) -equivariant retraction r H : X H → Γ OS ( G ) H . Then r H is a homology equivalence. Moreover, the stabilizer of the cellsin X H − Γ OS ( G ) H is H (the stabilizer cannot be bigger by the induction hypothesis). We defineretractions r H g : X H g → Γ OS ( G ) H g by r H g ( gx ) = g · r H ( x ) which glue to give a G -equivarianthomology equivalence r : Γ OS ( G ) (cid:91) g ∈ G X H g → Γ OS ( G ) . We may replace X by the pushout (cid:101) X given by the following diagram Γ OS ( G ) (cid:91) g ∈ G X H g Γ OS ( G ) X (cid:101) X rr IVÁN SADOFSCHI COSTA
It follows that r is a homology equivalence, so the resulting G -complex (cid:101) X is acyclic. Moreoversince (cid:101) X (1) is a subcomplex of X (1) and the restriction r : X (1) → (cid:101) X (1) is a retraction, r inducesan epimorphism on π . This procedure removes the excessive orbits of cells of type G/H . Byinduction we obtain a complex X (cid:48) such that X (cid:48) (1) coincides with Γ OS ( G ) up to k ≥ freeorbits of -cells and such that every orbit of -cells is free. By Lemma 2.6 X (cid:48) is an H-universal ( G, SLV ) -complex. Now by Lemma 2.10 and Proposition 2.11 there are exactly k + 1 orbits of -cells. (cid:3) From [OS02, Propositions 3.3 and 3.6] we have:
Proposition 3.7.
Each of the groups in the statement of Theorem 2.2 has a subgroup isomorphicto one of the following groups: • PSL (2 p ) for p prime; • PSL (3 p ) for an odd prime p ; • PSL ( q ) for a prime q > such that q ≡ ± and q ≡ ± ; • Sz(2 p ) for p an odd prime.Moreover, every proper subgroup of a group in this list is solvable. Theorem 3.8.
To prove Conjecture 1.1 it is enough to prove P ( G ) for each group G listed inProposition 3.7, where P ( G ) denotes the following proposition: “there is a nontrivial represen-tation in SO( n, R ) of the fundamental group of every acyclic G -complex obtained from Γ OS ( G ) by attaching k ≥ free orbits of -cells and ( k + 1) free orbits of -cells”.Proof. Let G be a finite group and suppose that X is a finite, acyclic -dimensional fixed pointfree G -complex. Let N be the subgroup generated by all normal subgroups N (cid:48) (cid:47) G such that X N (cid:48) (cid:54) = ∅ . By Theorem 2.3 we have that Y = X N is acyclic and the action of G/N on Y is essential and fixed point free. Then G/N must be one of the groups in Theorem 2.2. Wetake a subgroup K of G/N isomorphic to one of the groups listed in Proposition 3.7. Then byLemma 2.8 the action of K on Y is fixed point free. Now, by Theorem 3.6 and by P ( K ) , it followsthat π ( Y ) admits a nontrivial representation in SO( n, R ) . Therefore, by Proposition A.3, X cannot be contractible. (cid:3) Remark . In Theorem 7.4 we prove the group A ∼ = PSL (2 ) satisfies the condition P inTheorem 3.8.The following explains why our particular choice of Γ OS ( G ) and the way the free orbits of -cells are attached is not relevant. Proposition 3.10.
Any two choices for Γ OS ( G ) are G -homotopy equivalent. Moreover, attach-ing k ≥ free orbits of -cells to any two choices for Γ OS ( G ) produces G -homotopy equivalentgraphs.Proof. Since any choice of Γ OS ( G ) is a universal ( G, SLV − { } ) -complex, the first part followsfrom [OS02, Proposition A.6]. The second part follows easily from the first and the gluingtheorem for adjunction spaces [Bro06, 7.5.7]. (cid:3) Corollary 3.11.
Let Γ be a graph obtained from Γ OS ( G ) by attaching k ≥ free orbits of -cells.The set of G -homotopy equivalence classes of -dimensional acyclic fixed point free G -complexeswith -skeleton Γ does not depend on the particular choice of Γ OS ( G ) or the way the k free orbitsof -cells are attached. In particular, the set of isomorphism classes of groups that occur as thefundamental group of such spaces does not depend on such choices. ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 9 Proof.
Again, this is an easy application of [Bro06, 7.5.7]. (cid:3) Brown’s short exact sequence
Using Bass–Serre theory, K.S. Brown gave a method to produce a presentation for a group G acting on a simply connected complex X [Bro84, Theorem 1]. When X is not simply connected,Brown describes a presentation for an extension (cid:101) G X of G by π ( X ) [Bro84, Theorem 2]. Thegroup (cid:101) G X has a description as a quotient of the fundamental group of a graph of groups. Asimilar result in the simply connected case was given by Corson [Cor92, Theorem 5.1] in termsof complexes of groups (higher dimensional analogues of graphs of groups).Using Brown’s result we translate the A case of Conjecture 1.1 into a nice looking problemin combinatorial group theory. This translation can be done in general, but to obtain similarresults for the rest of the groups G that appear in Theorem 3.8 we need a choice of Γ OS ( G ) andpresentations for the stabilizers of its vertices.In Brown’s original formulation, the result deals with actions that need not to be admissible(Brown uses the term G − CW -complex in a different way than us). Since the actions we areinterested in are admissible, we state Brown’s result only in that case.Let X be a connected G -complex. By admissibility of the action, the group G acts on theset of oriented edges. If e is an oriented edge, the same -cell with the opposite orientation isdenoted by e − . Each oriented edge e has a source and target , denoted by s ( e ) and t ( e ) and forevery g ∈ G we have g · s ( e ) = s ( g · e ) and g · t ( e ) = t ( g · e ) .To obtain a description of the group (cid:101) G X we need a number of choices that we now specify.For each -cell of X we choose a preferred orientation in such a way that these orientations arepreserved by G . This determines a set P of oriented edges. We choose a tree of representatives for X/G . That is, a tree T ⊂ X such that the vertex set V of T is a set of representativesof X (0) /G . Such tree always exists and the -cells of T are inequivalent modulo G . We givean orientation to the -cells of T so that they are elements of P . We also choose a set ofrepresentatives E of P/G in such a way that s ( e ) ∈ V for every e ∈ E and such that eachoriented edge of T is in E . If e is an oriented edge, the unique element of V that is equivalentto t ( e ) modulo G will be denoted by w ( e ) . For every e ∈ E we fix an element g e ∈ G such that t ( e ) = g e · w ( e ) . If e ∈ T , we specifically choose g e = 1 . For each orbit of -cells we choose aclosed edge path τ based at a vertex of T and representing the attaching map for this orbit of -cells. Let F be the set given by these closed edge paths.The group (cid:101) G X is defined as a quotient of ∗ v ∈ V G v ∗ ∗ e ∈ E Z by certain relations. In order to define these relations we introduce some notation. If v ∈ V and g ∈ G v we denote the copy of g in the free factor G v by g v . The generator of the copy of Z thatcorresponds to e is denoted by x e . The relations are the following:(i) x e = 1 if e ∈ T .(ii) x − e g s ( e ) x e = ( g − e gg e ) w ( e ) for every e ∈ E and g ∈ G e .(iii) r τ = 1 for every τ ∈ F .We state Brown’s theorem before giving the definition of the element r ω associated to a closededge path ω . Theorem 4.1 (Brown, [Bro84, Theorems 1 and 2]) . The group (cid:101) G X = ∗ v ∈ V G v ∗ ∗ e ∈ E Z (cid:104)(cid:104) R (cid:105)(cid:105) where R consists of relations (i)-(iii) is an extension → π ( X, x ) i −→ (cid:101) G X φ −→ G → . The map φ is defined passing to the quotient the coproduct φ of the inclusions G v → G and themappings Z → G given by x e (cid:55)→ g e . The map i sends a closed edge path ω based at x ∈ V tothe class of r ω . Now we explain how to obtain the elements r ω . If α is an oriented edge, we define ε ( α ) = (cid:40) α ∈ P − if α / ∈ P and we can always take e ∈ E and g ∈ G such that α = ge ε ( α ) . Note that e is unique but g isnot. Moreover, if α starts at v ∈ V , we can write α = (cid:40) he with h ∈ G s ( e ) , if α ∈ Phg − e e − with h ∈ G w ( e ) , if α / ∈ P Again, h is not unique.Now if ω = ( α , . . . , α n ) is a closed edge path starting at a vertex v ∈ V we define theelement r ω ∈ ∗ v ∈ V G v ∗ ∗ e ∈ E Z . Recursively, we define some sequences. Since the oriented edge α starts at v ∈ V , we can obtain an oriented edge e and an element h ∈ G v as above.We set ε = ε ( α ) and g = h g ε e . Set v = w ( e ) if α ∈ P and otherwise v = s ( e ) . Nowsuppose we have defined e , . . . , e k , h , . . . , h k , ε , . . . , ε k , g , . . . , g k and v , . . . , v k such thatthe oriented edge ( g g · · · g k ) − α k +1 starts at v k ∈ V . We can obtain an oriented edge e k +1 and an element h k +1 ∈ G v k as before. We set ε k +1 = ε ( α k +1 ) and g k +1 = h k +1 g ε k +1 e k +1 . Set v k +1 = w ( e k +1 ) if α k +1 ∈ P and otherwise v k +1 = s ( e k +1 ) . When we conclude, we have anelement g g · · · g n ∈ G v . Finally the word associated to ω is given by r ω = ( h ) v x ε e ( h ) v x ε e · · · ( h n ) v n − x ε n e n ( g g · · · g n ) − v . A closed edge path ω in X determines a conjugacy class (cid:74) ω (cid:75) of π ( X ) . The following describesthe conjugation action of (cid:101) G X on π ( X ) . Proposition 4.2 ([Bro84, Proposition 1]) . Let ω be a closed edge path in X and g ∈ G . Then theconjugacy classes i ( (cid:74) ω (cid:75) ) and i ( (cid:74) gω (cid:75) ) are contained in the same (cid:101) G X -conjugacy class. Moreoverfor any element (cid:101) g ∈ φ − ( g ) we have (cid:101) gi ( (cid:74) ω (cid:75) ) (cid:101) g − = i ( (cid:74) gω (cid:75) ) . The following proposition summarizes many ideas of this section.
Proposition 4.3.
Let Γ be a G -graph and let w , . . . , w k ∈ ker( φ : (cid:101) G Γ → G ) . Let X be a G -complex obtained by attaching orbits of -cells to Γ along closed edge paths τ , . . . , τ k such that r τ i = w i . Then we have a diagram with exact rows and columns ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 11 (cid:104)(cid:104) G · τ i (cid:105)(cid:105) (cid:104)(cid:104) w i (cid:105)(cid:105) (cid:101) G Γ π (Γ) (cid:101) G Γ G π ( X ) (cid:101) G X G
11 1 1 i ∼ i ∗ i φi φ and we have H ( X ) ∼ = N (cid:104)(cid:104) w i (cid:105)(cid:105) (cid:101) G Γ [ N, N ] , where N = ker( φ : (cid:101) G Γ → G ) .Remark . If X is a connected G -complex, the group (cid:101) G X is isomorphic to the group formedby the pairs ( g, (cid:101) g ) such that g ∈ G and (cid:101) g is a lift of g : X → X to the universal cover (cid:101) X of X (see [Bro84]). Suppose Y is another G -complex and h : X → Y is equivariant and a homotopyequivalence. Let (cid:101) h : (cid:101) X → (cid:101) Y be a lift of h to the universal covers. Then if g ∈ G , for each lift (cid:101) g X : (cid:101) X → (cid:101) X of g : X → X there is a unique lift (cid:101) g Y : (cid:101) Y → (cid:101) Y of g : Y → Y such that the followingdiagram commutes: (cid:101) X (cid:101) Y (cid:101) X (cid:101) Y (cid:101) h (cid:101) g X (cid:101) g Y (cid:101) h Then it is easy to check that there is an isomorphism (cid:101) G X → (cid:101) G Y given by (cid:101) g X (cid:55)→ (cid:101) g Y . In particular,the isomorphism type of (cid:101) G Γ OS ( G ) does not depend on any choice.We now apply Brown’s result for G = A . Recall the construction of Γ OS ( A ) given in Defi-nition 2.16. Suppose that we have an acyclic -complex X obtained from Γ OS ( A ) by attachinga free A -orbit of -cells. We want to apply Brown’s method to obtain a presentation for theextension (cid:101) G X . We take T = { e , e } . Thus V = { v , v , v } . We take E = { e , e , e } .Note that we have w ( e ) = t ( e ) for every e ∈ E . We can take g e = 1 for every e ∈ E .Then Brown’s result gives (cid:101) G X = ( H ∗ H H ∗ H H ) ∗ H (cid:104)(cid:104) w (cid:105)(cid:105) We explain this. First we amalgamate the groups H , H , H identifying the copy of H in H with the copy of H in H and the copy of H in H with the copy of H in H . This comesfrom the relations of type (i) and (ii) for e ∈ T . Then we form an HNN extension with stableletter x = x e that corresponds to the relation of type (ii) coming from e . The associatedsubgroups of this HNN extension are the copies of H in H and H . The quotient by the word w comes from the only relation of type (iii).Now we obtain an explicit presentation for (cid:101) G X . We have (cid:104) a, b | a , b , ( ab ) (cid:105) ∼ = H via a (cid:55)→ (2 , , , b (cid:55)→ (3 , , . We have (cid:104) b, c | b , c , ( bc ) (cid:105) ∼ = H via b (cid:55)→ (3 , , , c (cid:55)→ (1 , , . Finally (cid:104) c, d | c , d , ( cd ) (cid:105) ∼ = H via c (cid:55)→ (1 , , , d (cid:55)→ (2 , , . Thus we have a presenta-tion (cid:101) G X = (cid:104) a, b, c, d, x | a , b , c , d , ( ab ) , ( bc ) , ( cd ) , xax − = d, w (cid:105) where the word w depends on the attaching map. The mapping φ : (cid:101) G X → A is given by a (cid:55)→ (2 , , , b (cid:55)→ (3 , , , c (cid:55)→ (1 , , , d (cid:55)→ (2 , , and x (cid:55)→ . Note that π ( X ) istrivial if and only if φ : (cid:101) G X → A is an isomorphism. If we also take into account k additionalfree orbits of and cells and we recall Theorem 3.6, from Brown’s result we obtain: Theorem 4.5.
The following are equivalent.(i) Every finite, -dimensional contractible A -complex has a fixed point.(ii) There is no presentation of A of the form (cid:104) a, b, c, d, x , . . . , x k | a , b , c , d , ( ab ) , ( bc ) , ( cd ) , x ax − = d, w , . . . , w k (cid:105) with w , . . . , w k ∈ ker( φ ) , where φ : F ( a, b, c, d, x , . . . , x k ) → A is given by a (cid:55)→ (2 , , , b (cid:55)→ (3 , , , c (cid:55)→ (1 , , , d (cid:55)→ (2 , , and x i (cid:55)→ . A moduli of representations
In order to prove Theorem 7.1 we define a moduli of representations of the group Γ k = (cid:104) a, b, c, d, x , . . . , x k | a , b , c , d , ( ab ) , ( bc ) , ( cd ) , x ax − = d (cid:105) in SO(3) . Our argument is inspired by James Howie’s proof of the Scott–Wiegold conjec-ture [How02].Let φ : Γ k → A be the homomorphism induced by φ : F ( a, b, c, d, x , . . . , x k ) → A . Lemma 5.1.
We have ker (cid:0) φ (cid:1) = (cid:104)(cid:104) x , . . . , x k , ( bac ) (cid:105)(cid:105) .Proof. It is straightforward to verify that ( bac ) ∈ ker( φ ) , so it is enough to show the inducedepimorphism φ : Γ k / (cid:104)(cid:104) x , . . . , x k , ( bac ) (cid:105)(cid:105) → A is in fact an isomorphism. Eliminating d and the x i we see Γ k / (cid:104)(cid:104) x , . . . , x k , ( bac ) (cid:105)(cid:105) = (cid:104) a, b, c | a , b , c , ( ab ) , ( bc ) , ( ca ) , ( bac ) (cid:105) . The quickest way to finish the proof is by using
GAP [GAP19] to compute the order of thisgroup: gap> F:=FreeGroup("a","b","c");;gap> AssignGeneratorVariables(F);;
In Appendix C we give an alternative proof by hand. (cid:3)
Proposition 5.2.
Let w , . . . , w k ∈ ker( φ ) . If the group Γ k admits a representation ρ such that(i) ρ ( w i ) = 1 for each i = 0 , . . . , k and(ii) there exists r ∈ { x , . . . , x k , ( bac ) } such that ρ ( r ) (cid:54) = 1 then Γ k / (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) φ −→ A is not an isomorphism.Proof. This follows from Lemma 5.1. (cid:3)
ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 13 Remark . Note that in some cases (for example when k = 0 and w = x ) a representation of Γ k with image isomorphic to A may suffice to conclude that Γ k / (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) is not A . Thismay seem counterintuitive.If α, β ∈ C we consider the matrix R ( α, β ) = α β − β α
00 0 1 which lies in
SO(3 , C ) whenever α + β = 1 . Recall that SO( n, C ) is the group of matrices M ∈ M n ( C ) such that M · M T = 1 and det( M ) = 1 . We now introduce our moduli ofrepresentations of Γ k . Theorem 5.4. If z = ( α , β , α , β , α , β , X , . . . , X k ) ∈ C × SO(3 , C ) k satisfies α i + β i = 1 for i = 1 , , then there is a group representation ρ z : Γ k → SO(3 , C ) defined by the following matrices A = − − √ − √ − B = − − √ √ −
00 0 1 C = R ( α , β ) S R ( α , β ) T D = R ( α , β ) S R ( α , β ) S R ( α , β ) T S T R ( α , β ) T X = R ( α , β ) S R ( α , β ) S R ( α , β ) S where S = − − , S = − − − , S = − cos( π ) 0 − sin( π )0 − − sin( π ) 0 cos( π ) , S = π ) sin( π )1 0 00 sin( π ) − cos( π ) and S = − √ − √ − √ √ .Proof. The proof reduces to the case k = 0 . We describe the computations needed to finishthe proof. It is straightforward to prove A = 1 , B = 1 and ( AB ) = 1 ; C = 1 and D = 1 reduce to S = 1 and S = 1 respectively. Since R ( α , β ) commutes with B , to prove ( BC ) = 1 it is enough to verify ( BS ) = 1 . To prove ( CD ) = 1 it is enough to verify that ( S T S S R ( α , β ) S R ( α , β ) T ) = 1 and, since S T S S = − − commutes with R ( α , β ) , this reduces to proving ( S T S S S ) = 1 which follows from S T S S S = cos( π ) 0 sin( π )0 1 0 − sin( π ) 0 cos( π ) . Finally, X AX T = D reduces to S R ( α , β ) S AS T R ( α , β ) T S T = S which follows from S AS T = S T S S = − − . The function check_rep in Appendix D gives an alternative proof using SAGE [Sag19]. (cid:3)
Remark . We also regard A , B , C , D , X as matrices with coefficients in the polynomial ring C [ α , β , α , β , α , β ] . Remark . This family of representations was obtained in the following way. We first obtaineda single representation of the group Γ in SO(3 , R ) by choosing reflections σ , σ , σ , σ , σ withaxes forming the appropriate angles so that a (cid:55)→ σ σ , b (cid:55)→ σ σ , c (cid:55)→ σ σ and d (cid:55)→ σ σ defines a representation of the (alternating Coxeter) group generated by a , b , c , and d . Since σ σ and σ σ are rotations of the same angle, they are conjugate, so it is possible to extendthis to a representation of Γ by mapping x to a rotation r . Then we twisted this represen-tation in the following way to obtain three degrees of freedom. If θ , θ and θ are rotationscommuting with σ σ , σ σ , and σ σ respectively then a (cid:55)→ σ σ , b (cid:55)→ σ σ , c (cid:55)→ θ σ σ θ − , d (cid:55)→ θ θ σ σ θ − θ − and x (cid:55)→ θ θ rθ gives a representation of Γ . After tidying up thesecomputations we obtain the moduli in Theorem 5.4. Remark . Given a family { w i } i ∈ I of words in F ( a, b, c, d, x , . . . , x k ) , the set of points z ∈ C × SO(3 , C ) k ⊆ C k such that ρ z ( w i ) = 1 for all i ∈ I is an affine algebraic variety that wedenote Z ( { w i : i ∈ I } ) . For k = 0 the variety Z ( w ) can be described with only equations.More generally, if we allow X , . . . , X k to take values in O(3 , C ) the variety Z ( w , . . . , w k ) canbe described using k equations. This suggests that it may be possible to use a result suchas Bézout’s theorem to count points. We could not finish this approach so we took a differentone. Proposition 5.8.
There is exactly one choice of ( α , β , α , β , α , β , X , . . . , X k ) ∈ C × SO(3 , C ) k with α i + β i = 1 for i = 1 , , such that the matrices in Theorem 5.4 satisfy X = X = . . . = X k = ( BAC ) = 1 . The unique solution z u = ( α u , β u , α u , β u , α u , β u , , . . . , is real and its exact value is given by α u = − (cid:112) √ α u = − (cid:113) − √ α u = − (cid:113) − √ β u = (cid:112) − √ β u = (cid:113) √ β u = (cid:113) √ . ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 15 Proof.
Again this reduces to the case k = 0 . In Appendix D we give a proof using SAGE. Weindicate here how to prove this by hand. We rewrite ( BAC ) = 1 as ( BAC ) − ( BAC ) T = 0 (i)and X = 1 as S R ( α , β ) S R ( α , β ) − R ( α , β ) T S T = 0 . (ii)We have BAC = − √ α β − ( α − β ) α β − √ ( α − β ) − √ − α β + √ ( α − β ) −√ α β − ( α − β ) − √ − √ α β − √ ( α − β ) . Then the (3 , entry of (i) gives √ α β + ( α − β ) − = 0 and from α + β = 1 we obtain √ α β + 89 α −
23 = 0 . (iii)To find the entries of (ii) it is useful to recall that cos( π ) = ( √ − , sin( π ) = (cid:112) √ , cos( π ) = ( √ , and sin( π ) = (cid:112) − √ . We obtain − √ α β − √ β + √ β − √ β β + √ α − α √ α + √ − √ β − α α − √ α − α β − β − β − √ − √ α − √ α β + √ β − √ − √ − √ β β − √ α √ α + √ = 0 . The (3 , entry determines the value of α . From (iii) we obtain the value of β . The (1 , entry allows to obtain the value of β . Now the (2 , entry gives the value of β . The (3 , entry gives the value of α and finally the (2 , entry determines the value of α . Computingthe remaining entries we see these values form a solution to X = ( BAC ) = 1 and satisfy α i + β i = 1 . (cid:3) Remark . We say that z u is universal in the following sense: if { w i } i ∈ I ⊆ ker( φ ) then z u ∈ Z ( { w i : i ∈ I } ) .The following result is proved in Section 6. Theorem 5.10.
Let w , . . . , w k ∈ ker( φ ) . Let N = ker( φ ) . If N = (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) Γ k [ N, N ] thenthe variety Z ( w , . . . , w k ) has at least two different points. Note that, by Proposition 4.3, the condition N = (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) Γ k [ N, N ] is equivalent to theacyclicity of the corresponding -complex. This is also the same as saying that w , . . . , w k generate the A -module N/ [ N, N ] (i.e. the relation module of → N → Γ k φ −→ A → ).6. Quaternions
To prove Theorem 5.10 we study the real part of the moduli, working with quaternions insteadof orthogonal matrices. This is useful because representing a rotation as a quaternion allows tofind the axis easily.Recall that S = { q ∈ H : | q | = 1 } acts on S = { b i + c j + d k : b + c + d = 1 } by conjugation. Recall that any element of S can be written as cos( θ/
2) + sin( θ/ q with θ ∈ [0 , π ] and q = b i + c j + d k ∈ S . There is a homomorphism p : S → SO(3 , R ) with ker( p ) = { , − } and which sends cos( θ ) + sin( θ )( b i + c j + d k ) to the rotation matrix with angle θ and axis ( b, c, d ) . Note that (cid:101) R ( t ) = cos( t ) + k sin( t ) is a lift of R (cos( t ) , sin( t )) by p . Let ψ : H → R be given by a + b i + c j + d k (cid:55)→ ( b, c, d ) . Recall that if q ∈ S and v is a purequaternion we have ψ ( qvq − ) = p ( q ) · ψ ( v ) . Let D ⊂ R be the unit disk. Let ϕ : D → H begiven by ( b, c, d ) (cid:55)→ √ − b − c − d + b i + c j + d k . We denote the coordinates of [0 , π ] × ( D ) k by t , t , t , . . . , t k +1) . Definition 6.1.
Let (cid:101) A , (cid:101) B , (cid:101) S , (cid:101) S , (cid:101) S , (cid:101) S , (cid:101) S , be preimages by p of the matrices A , B , S , S , S , S , S which appear in the statement of Theorem 5.4. We also define functions (cid:101) C, (cid:101) D, (cid:101) X : [0 , π ] × ( D ) k → H by (cid:101) C ( t ) = (cid:101) R ( t ) (cid:101) S (cid:101) R ( t ) − , (cid:101) D ( t ) = (cid:101) R ( t ) (cid:101) S (cid:101) R ( t ) (cid:101) S (cid:101) R ( t ) − (cid:101) S − (cid:101) R ( t ) − , (cid:101) X ( t ) = (cid:101) R ( t ) (cid:101) S (cid:101) R ( t ) (cid:101) S (cid:101) R ( t ) (cid:101) S . For i = 1 , . . . , k we define (cid:101) X i ( t ) = ϕ ( t i +1 , t i +2 , t i +3 ) . Let t u , t u , t u ∈ [0 , π ] be the uniquenumbers such that cos( t ui ) = α ui and sin( t ui ) = β ui . Let t u = ( t u , t u , t u , , . . . , ∈ [0 , π ] × ( D ) k .Note that we can arrange the signs of these preimages so that (cid:16) (cid:101) B (cid:101) A (cid:101) C (cid:17) ( t u ) = 1 and (cid:101) X ( t u ) = 1 .If w ∈ F ( a, b, c, d, x , . . . , x k ) there is an induced map (cid:102) W : [0 , π ] × ( D ) k → S . Note thatany two words w, w (cid:48) which are equal in Γ k induce maps (cid:102) W , (cid:102) W (cid:48) which are equal or differ on asign. If w , . . . , w k ∈ ker( φ ) we can consider (cid:102) W = ( (cid:102) W , . . . , (cid:102) W k ) : [0 , π ] × ( D ) k → ( S ) k +1 which can be composed with Ψ = ( ψ, . . . , ψ ) : H k +1 → R k +1) to obtain a map Ψ (cid:102) W : [0 , π ] × ( D ) k → ( D ) k +1 . The plan is to assume t u is the only zero in order to do a degree argument. We will get acontradiction by computing the degree in two different ways. Lemma 6.2.
Let I = [ − , and let D ⊂ R be the unit disk. Let F = ( f , . . . , f k ) : I × ( D ) k → ( D ) k +1 be a continuous map which is nonzero on the boundary of I × ( D ) k and satisfies the followingparity condition: • For t , t , t ∈ I , x , . . . , x k ∈ D we have ( f , f , . . . , f k )(( − , t , t ) , x , . . . , x k ) = ( − f , f , . . . , f k )((1 , t , t ) , x , . . . , x k )( f , f , . . . , f k )(( t , − , t ) , x , . . . , x k ) = ( − f , f , . . . , f k )(( t , , t ) , x , . . . , x k )( f , f , . . . , f k )(( t , t , − , x , . . . , x k ) = ( − f , f , . . . , f k )(( t , t , , x , . . . , x k ) . • For each ≤ i ≤ k and for every ( x , . . . , x k ) ∈ I × ( D ) k with x i ∈ ∂ D we have ( f , f , . . . , f k )( x , . . . , x i − , − x i , x i +1 , . . . , x k ) = ( f , . . . , f i − , − f i , f i +1 , . . . , f k )( x , . . . , x k ) . Then the restriction F : ∂ ( I × ( D ) k ) → ( D ) k +1 − { } has even degree. ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 17 Proof.
We fix cellular structures. For I we take the structure with two -cells and one -cell.For D we take the cell structure with two -cells, two -cells, two -cells and one -cell (theantipodal map interchanges the i -cells in each pair for ≤ i ≤ ). We take the product cellularstructure for I , I × ( D ) k and ( D ) k +1 . Let S = ∂ ( I × ( D ) k ) . Note that the (3 k + 2) -cellsof S can be divided into k pairs of opposite cells in a natural way. Note that it is easy todefine a cellular map h : I → ∂ D which satisfies h ( − , t , t ) = − h (1 , t , t ) h ( t , − , t ) = − h ( t , , t ) h ( t , t , −
1) = − h ( t , t , . Let h i : D → D be the identity for ≤ i ≤ k . Now we can define a homotopy between F | S anda map G : S → ∂ ( D ) k +1 that satisfies the parity condition and coincides with H = ( h , . . . , h k ) on the (3 k + 1) -skeleton of S . This is done skeleton by skeleton using that ∂ ( D ) k +1 is (3 k + 1) -connected. For each pair of opposite (3 k + 2) -cells we can extend the homotopy so that theparity condition is also satisfied by G . Clearly the degrees of F | S and G are equal. Now notethat if e, e (cid:48) is a pair of opposite (3 k + 2) -cells then G ∗ ( e ) , H ∗ ( e ) ∈ C k +2 ( ∂ ( D ) k +1 ) differ on anelement of H k +2 ( ∂ ( D ) k +1 ) . Moreover, by the parity condition, G ∗ ( e (cid:48) ) and H ∗ ( e (cid:48) ) differ on thesame element. Thus the degree of H | S and the degree of G are equal modulo . To conclude,note that deg( H | S ) = 0 since H : I × ( D ) k → ∂ ( D ) k +1 is an extension of H | S to a contractiblespace. (cid:3) Corollary 6.3.
Let w , . . . , w k ∈ F ( a, b, c, d, x , . . . , x k ) be words and assume the total exponentof x i in w j is δ i,j . If Ψ (cid:102) W is nonzero on the boundary of [0 , π ] × ( D ) k , then the degree of therestriction Ψ (cid:102) W : ∂ (cid:0) [0 , π ] × ( D ) k (cid:1) → ( D ) k +1 − { } is even.Proof. Since the total exponent of x i in w j is δ i,j , by looking at Definition 6.1 we see the paritycondition of Lemma 6.2 is satisfied. (cid:3) Recall that the degree can be computed in the following way
Lemma 6.4.
Let f : R n → R n be smooth and assume f (0) = 0 . If det( Df ) (cid:54) = 0 then is anisolated zero and the degree of f around is given by deg( f,
0) = sg(det( Df )) . We need some basic differentiation properties for quaternion valued analytic functions analo-gous to the usual ones (see Appendix B). Note that (cid:101) R ( t ) = cos( t ) + k sin( t ) = 1 + t k + O ( t ) . Lemma 6.5.
Let (cid:101) X = ( (cid:101) X , . . . , (cid:101) X k ) . Then D (cid:16) Ψ (cid:101) X (cid:17) t u is invertible.Proof. Again this reduces to the case k = 0 by noting that D (cid:16) Ψ (cid:101) X (cid:17) t u = (cid:32) M I (cid:33) where M is the × matrix we obtain in the k = 0 case. We now prove M is invertible. Recallthat (cid:101) X ( t u ) = 1 . Then (cid:101) X ( t u + t ) = (cid:101) R ( t u ) (cid:101) R ( t ) (cid:101) S (cid:101) R ( t u ) (cid:101) R ( t ) (cid:101) S (cid:101) R ( t u ) (cid:101) R ( t ) (cid:101) S = (cid:101) R ( t u ) (cid:18) t k (cid:19) (cid:101) S (cid:101) R ( t u ) (cid:18) t k (cid:19) (cid:101) S (cid:101) R ( t u ) (cid:18) t k (cid:19) (cid:101) S + O ( t )= 1 + 12 (cid:101) R ( t u ) k (cid:101) R ( t u ) − t + 12 (cid:16) (cid:101) R ( t u ) (cid:101) S (cid:101) R ( t u ) (cid:17) k (cid:16) (cid:101) R ( t u ) (cid:101) S (cid:101) R ( t u ) (cid:17) − t + 12 (cid:101) S − k (cid:101) S t + O ( t ) Now recalling that q k q − = ( i , j , k ) · p ( q ) · (0 , , for any q ∈ S we see that the columns of M are given by R ( α u , β u ) · (0 , ,
1) = (cid:18) , , (cid:19) R ( α u , β u ) S R ( α u , β u ) · (0 , ,
1) = (cid:18) − β u , − α u , (cid:19) S − · (0 , ,
1) = (cid:18) , − √ , √ (cid:19) . Thus M = − β u − α u − √ √ and therefore det( M ) = √ β u (cid:54) = 0 . (cid:3) Lemma 6.6.
Let w ∈ ker( φ ) . Then ∂ (cid:102) W∂t i ( t u ) is a pure quaternion for i = 1 , . . . , k + 1) .Proof. Since w belongs to ker( φ ) , in Γ k it equals a product of conjugates of the x i , ( bac ) andtheir inverses. Recall that S is invariant by the action of S . By Proposition B.1, it is enoughto prove that ∂ (cid:101) X j ∂t i ( t u ) and ∂ ( (cid:101) B (cid:101) A (cid:101) C ) ∂t i ( t u ) are pure quaternions.For i = 0 the first claim follows from the computation in the proof of Lemma 6.5 and is easyto verify for i > . The second claim follows similarly by noting that (cid:16) (cid:101) B (cid:101) A (cid:101) C (cid:17) ( t u ) = 1 andwriting (cid:16) (cid:101) B (cid:101) A (cid:101) C (cid:17) ( t u + t ) = (cid:18) (cid:101) B (cid:101) A (cid:18) t k (cid:19) (cid:101) S (cid:18) − t k (cid:19)(cid:19) + O ( t ) . (cid:3) Lemma 6.7.
Let N = ker( φ ) and let w , . . . , w k ∈ ker( φ ) . If N = (cid:104)(cid:104) w , . . . , w k (cid:105)(cid:105) Γ k [ N, N ] then D (cid:16) Ψ (cid:102) W (cid:17) t u is invertible.Proof. We may assume without loss of generality that (cid:102) W j ( t u ) = 1 for all j . For each j thereare numbers a j , (cid:96) j ∈ N , words v j, , . . . , v j,a j , u j, , . . . , v j,a j ∈ ker( φ ) , words p j, , . . . p j,(cid:96) j ∈ F ( a, b, c, d, x , . . . , x k ) , indices α j, , . . . , α j,(cid:96) j ∈ { , . . . , k } and signs ε j, , . . . , ε j,(cid:96) j ∈ { , − } suchthat in Γ k we have x j = (cid:96) j (cid:89) s =1 p j,s w ε j,s α j,s p − j,s a j (cid:89) i =1 [ u j,i , v j,i ] . ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 19 Then we have (cid:101) X j ( t u + t ) = (cid:96) j (cid:89) s =1 (cid:101) P j,s (cid:102) W ε j,s α j,s (cid:101) P − j,s a j (cid:89) i =1 [ (cid:101) U j,i , (cid:101) V j,i ] ( t u + t ) and using Proposition B.1 we obtain ∂ (cid:101) X j ∂t i ( t u ) = (cid:96) j (cid:88) s =1 (cid:101) P j,s ( t u ) ∂ (cid:102) W ε j,s α j,s ∂t i ( t u ) (cid:101) P − j,s ( t u )= (cid:96) j (cid:88) s =1 ε j,s (cid:101) P j,s ( t u ) ∂ (cid:102) W α j,s ∂t i ( t u ) (cid:101) P − j,s ( t u ) By Lemma 6.6, D (cid:16) Ψ (cid:102) W (cid:17) t u is invertible if and only if (cid:40)(cid:32) ∂ (cid:102) W ∂t i ( t u ) , . . . , ∂ (cid:102) W k ∂t i ( t u ) (cid:33) : 1 ≤ i ≤ k + 1) (cid:41) is linearly independent over R . If λ i ∈ R satisfy k +1) (cid:88) i =1 λ i ∂ (cid:102) W ∂t i ( t u ) = 0 it follows that k +1) (cid:88) i =1 λ i ∂ (cid:101) X ∂t i ( t u ) = 0 . By Lemma 6.5, D (cid:16) Ψ (cid:101) X (cid:17) t u is invertible and, again by Lemma 6.6, the set (cid:40)(cid:32) ∂ (cid:101) X ∂t i ( t u ) , . . . , ∂ (cid:101) X k ∂t i ( t u ) (cid:33) : 1 ≤ i ≤ k + 1) (cid:41) is linearly independent over R . Thus λ = . . . = λ k +1) = 0 and we are done. (cid:3) Proof of Theorem 5.10.
We can assume that the total exponent of x i in w j is δ i,j . To prove this,consider the abelianization and note that it is possible to achieve this by using the followingoperations: • replacing w i by w i w j (if i (cid:54) = j ), • replacing w i by w − i , and • interchanging w i and w j .By Lemma 6.7 and Lemma 6.4, the degree of Ψ (cid:102) W near t u is ± . If Ψ (cid:102) W has a zero on ∂ ([0 , π ] × ( D ) k ) we are done. Otherwise, by Corollary 6.3, the degree of Ψ (cid:102) W restricted tothe boundary of [0 , π ] × ( D ) k is even. It follows that there must be a point t (cid:54) = t u such that Ψ (cid:102) W ( t ) = 0 . This gives a second point in Z ( w , . . . , w k ) . (cid:3) Group actions of A on contractible -complexes We can now prove the following.
Theorem 7.1.
There is no presentation of A of the form (cid:104) a, b, c, d, x , . . . , x k | a , b , c , d , ( ab ) , ( bc ) , ( cd ) , x ax − = d, w , . . . , w k (cid:105) with w , . . . , w k ∈ ker( φ ) , where φ : F ( a, b, c, d, x , . . . , x k ) → A is given by a (cid:55)→ (2 , , , b (cid:55)→ (3 , , , c (cid:55)→ (1 , , , d (cid:55)→ (2 , , and x i (cid:55)→ for each i = 0 , . . . , k . Proof.
This follows from Theorem 5.10, Proposition 5.8 and Proposition 5.2. (cid:3)
Now from Theorem 7.1 and Theorem 4.5 we deduce.
Theorem 7.2.
Every action of A ∼ = PSL (2 ) on a finite, contractible -complex has a fixedpoint. Corollary 7.3.
Let G be one of the groups PSL (2 k ) , PSL (5 k ) for k ≥ or PSL ( q ) for q ≡ ± and q ≡ ± . Then every action of G on a finite contractible -complex has afixed point.Proof. Let G be one of these groups and let X be a finite acyclic fixed point free -dimensional G -complex. By [OS02, Proposition 3.3], A is a subgroup of G and by Lemma 2.8 the action of A on X is fixed point free. By Theorem 7.2 X cannot be contractible. (cid:3) Looking more carefully at the proof of Theorem 7.2 we obtain the following.
Theorem 7.4.
Let X be a fixed point free -dimensional finite, acyclic A -complex. Then thereis a nontrivial representation π ( X ) → SO(3 , R ) .Proof. By Theorem 3.6 we see that π surjects onto the fundamental group of an acyclic -dimensional A complex X (cid:48) which is obtained from Γ OS ( A ) by attaching k ≥ free orbits of -cells and k + 1 free orbits of -cells. Now note that the representation constructed to proveTheorem 7.1 restricted to π ( X (cid:48) ) gives a nontrivial morphism into SO(3 , R ) . (cid:3) Since A is the only finite perfect subgroup of SO(3 , R ) we deduce the following. Corollary 7.5.
Let X be a fixed point free -dimensional finite and acyclic A -complex and let π = π ( X ) . Then π is infinite or there is an epimorphism π → A . Recall that N = ker (cid:0) φ (cid:1) is a free group of rank k + 1) . We can restate Theorem 7.1 in thefollowing way which highlights the connection with the relation gap problem (see [Har18, Har15]). Corollary 7.6.
The extension → N → Γ k φ −→ A → has a relation gap. That is, the A -module N/ [ N, N ] is free of rank k + 1 . However N cannotbe generated by k + 1 elements as a Γ k -group. Note that since Γ k is not free this is not an example of a presentation with a relation gap. Appendix A. Equations over groups
Let G be a group. An equation over G in the variables x , . . . , x n is an element w ∈ G ∗ F ( x , . . . , x n ) . We say that a system of equations w ( x , . . . , x n ) = 1 w ( x , . . . , x n ) = 1 · · · w m ( x , . . . , x n ) = 1 has a solution in an overgroup of G if the map G → G ∗ F ( x , . . . , x m ) / (cid:104)(cid:104) w , . . . , w m (cid:105)(cid:105) is injective.Such a system of equations determines an ( m × n ) -matrix M where M i,j is given by the total ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 21 exponent of the letter x j in the word w i . A system is said to be independent if the rank of M is m .One of the most important open problems in the theory of equations over groups is theKervaire–Laudenbach–Howie conjecture [How81, Conjecture]. Conjecture A.1 (Kervaire–Laudenbach–Howie) . An independent system of equations over G has a solution in an overgroup of G . The Gerstenhaber–Rothaus theorem [GR62, Theorem 3] says that finitely generated sub-groups of compact connected Lie groups satisfy Conjecture A.1.
Proposition A.2.
Let X be a finite acyclic -complex and let A ⊂ X be an acyclic subcomplex.Then we can write π ( X ) = π ( A ) ∗ F ( x , . . . , x n ) / (cid:104)(cid:104) w , . . . , w n (cid:105)(cid:105) and the ( n × n ) -matrix M such that M i,j is the total exponent of x j in w i is invertible.Proof. Take a maximal tree T for A and consider a maximal tree T of X containing T . Then A/T (cid:39) A is an acyclic subcomplex of the acyclic -complex X/T (cid:39) X . As usual, from A/T we can read a presentation for π ( A ) which is balanced since A/T is acyclic. Now we considera variable x i for each -cell of X/T which is not in
A/T and we read words from the attachingmaps for the -cells of X/T which are not part of
A/T . In this way we obtain equations inthese variables with coefficients in A which give the desired description of π ( X ) . Since X/T isacyclic, there is an equal number of variables and equations and the matrix M is invertible. (cid:3) Now from the Gerstenhaber–Rothaus theorem we deduce.
Proposition A.3.
Let X be a finite acyclic -complex. If A ⊂ X is an acyclic subcomplex andthere is a nontrivial representation ρ : π ( A ) → SO( n, R ) then π ( X ) is nontrivial.Proof. By Proposition A.2 can write π ( X ) = π ( A ) ∗ F ( x , . . . , x n ) / (cid:104)(cid:104) w , . . . , w n (cid:105)(cid:105) and thesystem is independent. Let G = ρ ( π ( A )) . There is an induced map ρ : π ( A ) ∗ F ( x , . . . , x n ) → G ∗ F ( x , . . . , x n ) which induces an epimorphism π ( X ) → G ∗ F ( x , . . . , x n ) / (cid:104)(cid:104) ρ ( w ) , . . . , ρ ( w n ) (cid:105)(cid:105) . Finally from[GR62, Theorem 3] it follows that this group is nontrivial. (cid:3) Appendix B. Quaternion valued analytic functions A quaternion valued analytic function is a function f : U → H where U ⊂ R n is open, suchthat its components are analytic, that is a function that can be written as f = f + f i i + f j j + f k k with f , f i , f j , f k : U → R are analytic. For i = 1 , . . . , n we can define the partial derivative ∂f∂t i = ∂f ∂t i + ∂f i ∂t i i + ∂f j ∂t i j + ∂f k ∂t i k . We define Df t = (cid:18) ∂f∂t ( t ) , . . . , ∂f∂t n ( t ) (cid:19) . If each coordinate of F = ( f , . . . , f m ) : U → H m is analytic then we use the notation ∂ F ∂t i = (cid:18) ∂f ∂t i , . . . , ∂f m ∂t i (cid:19) . The usual properties extend to this context. We need the following
Proposition B.1.
Let f, g : U → H be analytic. Then(i) We have the product rule ∂f · g∂t i ( t ) = ∂f∂t i ( t ) g ( t ) + f ( t ) ∂g∂t i ( t ) . (ii) Suppose f is nowhere zero and g ( t ) ∈ R then ∂ f · g · f∂t i ( t ) = f ( t ) ∂g∂t i ( t ) f ( t ) − . (iii) Suppose f ( t ) = ± then ∂ f ∂t i ( t ) = − ∂f∂t i ( t ) . (iv) Suppose that f, g are nowhere zero and f ( t ) , g ( t ) ∈ { , − } . Then the commutator [ f, g ] = f · g · f · g satisfies [ f, g ]( t ) = 1 and ∂ [ f, g ] ∂t i ( t ) = 0 . Proof. (i) is a straightforward computation, (ii) and (iii) follow from (i). Finally, (iv) followsfrom the previous properties. (cid:3)
As usual we have the Taylor series f ( t + t ) = f ( t ) + n (cid:88) i =1 ∂f∂t i ( t ) t i + O ( t ) . From the product rule we see that we can multiply the Taylor series of two functions to obtainthe Taylor series of the product.
Appendix C. Alternative proof of Lemma 5.1
An alternative way to finish the proof of Lemma 5.1 goes by noting that x = bc and y = ca satisfy x = y = ( xy ) = 1 . Thus it would suffice to to show the group (cid:104) a, b, c | a , b , c , ( ab ) , ( bc ) , ( ca ) , ( bac ) (cid:105) ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 23 is generated by x, y , for it is well known that (cid:104) x, y | x , y , ( xy ) (cid:105) is a presentation of A . To dothis, it is enough to show that a ∈ (cid:104) x, y (cid:105) . The following computation proves this claim. xy xy − xy = ( bc )( ca )( ca )( bc )( a − c − )( a − c − )( bc )( ca ) (using a = c = 1 ) = bacabcacacba (replacing acba by b − cab − c ) = bacabcacb − cab − c (replacing cb − c by b ) = bacabcabab − c (replacing aba by b ab ) = bacabcb abc (replacing bcb by c ) = bacacbabc (replacing acba by b − cab − c ) = bacb − cab − cbc (replacing cb − c by b ) = babab − cbc (replacing cbc by b − ) = babab − (using b = 1 ) = babab (using ( ab ) = 1 ) = a. Appendix D. Alternative proofs using SAGE
The following SAGE code gives alternative proofs of Theorem 5.4 and Proposition 5.8 whichare easier to verify. Note that SAGE computes exactly over the algebraic numbers so there isno numerical error. The function check_rep , shows
A, B, C, D, X satisfy the defining relationsfor Γ in M ( C [ α , β , α , β , α , β ] / (cid:104) α + β − , α + β − , α + β − (cid:105) ) . The function find_universal_representations gives the exact value of the unique solution z u of X = ( BAC ) = 1 by solving the corresponding system of polynomial equations over thealgebraic closure of Q . Note that in this code we use x i , y i instead of α i , β i . def R(x,y):return matrix([(x,y,0,),(-y,x,0),(0,0,1),]);A = matrix([(-1,0,0),(0,1/3,-2/3*sqrt(2)),(0,-2/3*sqrt(2),-1/3),]);B = matrix([(-1/2,-sqrt(3)/2,0),(sqrt(3)/2, -1/2,0),(0,0,1),]); S0 = matrix([(-1,0,0),(0,1,0),(0,0,-1)])S1 = matrix([(-1,0,0),(0,0,-1),(0,-1,0),]);S2 = matrix([(-cos(2*pi/5),0,-sin(2*pi/5)),(0,-1,0),(-sin(2*pi/5),0,cos(2*pi/5)),]);S3 = matrix([(0,cos(pi/5),sin(pi/5)),(1,0,0),(0,sin(pi/5),-cos(pi/5)),]);S4 = matrix([(0, -sqrt(3)/3, -sqrt(6)/3),(1,0,0),(0,-sqrt(6)/3,sqrt(3)/3),]);def rep(x1,y1,x2,y2,x3,y3):C = R(x1,y1) * S0 * R(x1,y1).T;D = R(x1,y1) * S1 * R(x2,y2) * S2 * R(x2,y2).T * S1.T * R(x1,y1).T;X0 = R(x1,y1) * S1 * R(x2,y2) * S3 * R(x3,y3) * S4;return (A,B,C,D,X0);I = matrix.identity(3);def check_rep():R.
ROUP ACTIONS OF A ON CONTRACTIBLE -COMPLEXES 25 M3f = M3R.hom(f,M3S);
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Departamento de Matemática - IMAS, FCEyN, Universidad de Buenos Aires. Buenos Aires,Argentina.
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