Seminaire BOURBAKI Novembre 2001 54e annee, 2001-2002, n° 894, p. 1 a 17 FINITE GROUP ACTIONS ON ACYCLIC 2-COMPLEXES by Alejandro ADEM 1. A BRIEF HISTORY AND MOTIVATION A simple consequence of the Brouwer fixed point theorem is that any cyclic group acting on a closed disk IDn must have a fixed point. The classical work of P.A. Smith [18] shows that if P is a finite p-group, then any action of P on ~n must have a fixed point. From this there arises a very evident question: is there a group of compose order which can act on some ~n without any fixed points? This was settled in the affirmative by Floyd and Richardson in 1959 (see [7]), when they constructed fixed point free actions of the alternating group A5 on disks. These examples stood out as special exceptions for several years - indeed no other such actions were known to exist until Oliver (see [13]) obtained a complete characterization of those finite groups which can act on disks without stationary points. To explain it we first need to introduce some group-theoretic concepts. DEFINITION 1.1. - For p and q primes, let ~p be the class of finite groups G with normal subgroups P a H a G, such that P is of p-power order, G/H is of q-power order, and H/P is cyclic; and let Gp ~qGqp, G ~pGp. = = THEOREM 1.2. - A finite group G has a fixed point free action on a disk if and only if G / 9. In particular, any non-solvable group has a fixed point free action on a disk, and an abelian group has such an action if and only if it has three or more non-cyclic Sylow subgroups. The smallest group with a fixed point free action on a disk is in fact the alternating group A5; the smallest abelian group with such an action is C30 x C3o. Oliver also proved that a group G will have a fixed point free action on a finite Zp-acyclic(l) ~~ Recall that a complex X is said to be Zp -acyclic if its reduced mod p homology is identically zero; if its reduced integral homology vanishes it is said to be acyclic. 2 complex if and only if G ~ Note that a group G will act without fixed points on a contractible complex if and only if it acts without fixed points on an acyclic complex. Taking into account Oliver’s result, an obvious problem is that of constructing fixed point free actions on contractible or acyclic complexes of small dimension. A well- known theorem by J.-P. Serre states that any finite group acting on a tree must have a fixed point (see [17]). However, the situation for contractible 2-dimensional complexes is much more complicated - in fact it is an open question whether or not it is possible for a finite group to act on such a complex without fixed points. We will restrict our attention from now on to the case of acyclic 2-dimensional complexes. Our starting point is the classical example of an A5-action on an acyclic 2-dimensio- nal complex without fixed points, which we now briefly recall. In fact it is an essential ingredient in the construction due to Floyd and Richardson which we discussed above. This example is constructed by considering the left A5 action on the Poincare sphere ~3 = SO(3)/ A5 ; as the action has a single fixed point (corresponding to the fact that A5 is self-normalizing in SO(3)) we may remove an open 3-disk U around it to obtain an acyclic compact 3-manifold ~3 - U with a fixed point free action of A5. This in turn can be collapsed to a 2-dimensional subcomplex X ~ E3 - U upon which A5 still acts without fixed points. Equivalently we could identify E3 with the space obtained by identifying opposite faces of the solid dodecahedron in an appropriate way(2) and consider the A5 action induced by the usual action on the dodecahedron. The fixed point is the center of D and by collapsing to its boundary we obtain an explicit 2- dimensional complex X = with a fixed point free action of A5 which has 6 pentagonal 2-cells, 10 edges and 5 vertices. Note that if we take the join A = A5 *X with the induced diagonal action of A5, then we obtain a simply connected and acyclic complex, hence a contractible complex with a fixed point free action. From this we can obtain a fixed point free A5 action on a disk via regular neighborhoods (as explained in [4], p.57). This is the basic step in the construction of the Floyd-Richardson examples. Now an obvious question arises from all of this: can we characterize those finite groups which can act without fixed points on acyclic 2-dimensional complexes? In- deed, are there even other examples of such actions? Remarkably it turns out that these actions are only possible for a small class of simple groups, and their precise determination and description will require using the classification of finite simple groups. (2)To be precise: identify opposite faces of the dodecahedron by the map which pushes each face through the dodecahedron and twists it by 27T/10 about the axis of the push in the direction of a right hand screw (see [12]). 2. STATEMENT OF RESULTS In this note we will report on recent work of Oliver and Segev (see [15]) where they provide a complete description of the finite groups which can act on a 2-dimensional acyclic complex without fixed points. Their work builds on previous contributions by Oliver [13], [14], Segev [16] and Aschbacher-Segev [2]. To state their main result we need to introduce a useful technical condition for G-CW complexes. From now on we will use the term G-complex to refer to a G-CW-complex, however these results also hold for simplicial complexes with an admissible G-action(3). DEFINITION 2.1. - A G-complex X is said to be essential if there is no normal subgroup 1 ~ N a G with the property that for each H C G, the inclusion X H N X H induces an isomorphism on integral homology. If there were such a normal subgroup N, then the G-action on X is ’essentially’ the same as the G-action on X N, which factors through a G/N-action. For 2-dimensional complexes we have: THEOREM 2.2. - Let G be any finite group and let X be any 2-dimensional acyclic G-complex. Let N denote the subgroup generated by all normal subgroups N’ a G such that X N’ =1= 0. Then X N is acyclic, X is essential if and only if N =1, and if N ~ 1 then the action of G/N on X N is essential. Based on this we restrict our attention to essential complexes, and we can now state the main result in [15] : THEOREM 2.3. - Given a finite group G, there is an essential fixed point free 2-dimensional acyclic G-complex if and only if G is isomorphic to one of the simple groups for l~ > 2, PSL2 (q) for q - ~3 (mod 8) and q > 5 or for odd k 3. Furthermore the isotropy subgroups of any such G-complex are all solvable. Among the groups listed above, only the Suzuki groups Sz(q) are not commonly known; we will provide a precise definition for them as subgroups of GL4(Fq) in § 5. Note that the theorem is stated for arbitrary acyclic 2-dimensional complexes; there is no need to require that the complexes be finite. Our main goal will be to explain the proof of this result. This naturally breaks up into a number of different steps. We begin in § 3 by explaining how the theorem can be reduced to simple groups, based mostly on a theorem due to Segev [16]. Next in § 4 we describe techniques for constructing the desired actions, using methods derived from Oliver’s original work on group actions on acyclic complexes as well as a more detailed analysis of the associated subgroup lattices. This is then applied in § 5 to (3) A simplicial complex X with a G action is called admissible if the action permutes the simplices linearly and sends a simplex to itself only via the identity. provide explicit descriptions of fixed point free actions on an acyclic 2-complex for the simple groups listed in the main theorem. In §6 we sketch conditions which imply the non-existence of fixed point free actions on acyclic 2-complexes for most simple groups; this part requires detailed information about the intricate subgroup structure for the finite simple groups. Finally in § 7 we use the classification of finite simple groups and the previous results to outline the proof of the main theorem, which has been previously reduced to verification for simple groups. We also make a few concluding remarks. Remark 2.4. The background required to understand these results and their proofs - includes: (1) very basic equivariant algebraic topology; (2) familiarity with subgroup complexes and related constructions; and (3) a very detailed knowledge of the sub- group structure of the finite simple groups. As many of the arguments in the proofs depend on the particular properties of these groups, our synopsis cannot hope to contain complete details. However the original paper by Oliver and Segev [15] is writ- ten in a clear style accessible to a broad range of mathematicians and hence those interested in a deeper understanding of the results presented here should consult it directly. 3. REDUCTION TO SIMPLE GROUPS The goal of this section will be to explain how we can restrict our attention to finite simple groups. This is based on the following key result due to Segev [16] : THEOREM 3.1. - Let X be any 2-dimensional acyclic G-complex. Then the sub- complex of fixed points X G is either acyclic or empty. If G is solvable then X G is acyclic. Proof. - Although Segev’s original proof uses the Odd Order Theorem, it can be proved more directly. One can show that if X is an acyclic G-complex, then = H2(XC,Z) = 0. Hence we are reduced to establishing that there is only one connected component (provided XC is non-empty). For solvable groups this can be proved directly using induction and Smith Theory. Otherwise we consider a minimal group G for which a counterexample exists. If XC has k components then in fact it can be shown that X looks roughly like the join of an acyclic fixed point free G-complex Y with a set of k points. However as X is 2-dimensional, Y would have to be 1-dimensional, in other words a tree, and this cannot hold. 0 Remark 3. 2. - The reader should keep in mind that Theorem 3.1 is a basic tool in many of our subsequent arguments and it will be used explicitly and implicitly on several occasions. COROLLARY 3.3. -Let X be any 2-dimensional acyclic G-complex. Assume that A, B C X are G-invariant acyclic subcomplexes such that X G C A U B; then Proof. Assume that A n B = 0 and let Z denote the G-complex obtained by - identifying A and B each to a point. Then Z is acyclic since A and B both are, and ZG consists of two points, thus contradicting Theorem 3.1. D As an immediate consequence of Corollary 3.3 we obtain LEMMA 3.4. Let X be a 2-dimensional acyclic G-complex. Then if H, K C G are - such that H C NG (K) and X H, X K are both nou-empty, then ~. Moreover, if H C G is such that X H = ~, then 0. Proof. - Since H normalizes K, both XH and X~ are H-invariant acyclic subcom- plexes of X. Hence we conclude from Corollary 3.3 that ~ ~ X H n X K = X H K. For the second part, it suffices to prove it when H is minimal among subgroups without fixed points. Fix a pair M, M’ C H of distinct maximal subgroups (note that by Theorem 3.1, H is non-solvable). Then X M and XM’ are non-empty, but X M ~ XM’ = XM,M’~ = XH = 0. Hence X M and XM’ are disjoint CG (H)- invariant acyclic subcomplexes of X, meaning (by Corollary 3.3) that their union cannot contain whence it must be non-empty. 0 We can now prove one of the main reduction results, which allows us to restrict our attention to essential complexes. THEOREM 3.5. - Let G be any finite group, and let X be any 2-dimensional acyclic G-complex. Let N be the subgroup generated by all normal subgroups N’ a G such that X N’ =1= 0. Then X N is acyclic; X is essential if and only if N =1 and if N ~ 1 then the action of G/N on X N is essential. Proof If X Nl ~ ~ and X N2 ~ ~ for Ni , N2 G, then X(Nl,N2) 7~ 0 by Lemma - 3.4. So we infer that X N is non-empty, hence acyclic (by Theorem 3.1). Note that the action of any non-trivial normal subgroup of G/N on X N has empty fixed point set, hence the action of G/N on X N is always essential. Finally, assume that ~V ~ 1; by Theorem 3.1 we have that for all H C G, X H and X NH are acyclic or empty; and X NH ~ ~ if X~ ~ 0, by Lemma 3.4. Hence the inclusion X H is always an equivalence of integral homology, and hence X is not essential. D This result will allow us to focus our attention on actions of simple groups. THEOREM 3.6. - If G is a non-trivial finite group for which there exists an essential 2-dimensional acyclic G-complex X, then G is almost simple. In fact there is a normal simple subgroup L a G such that X L = ~ and such that CG (L) =1. Proof. - We know from Theorem 3.5 that XN = 0 for all normal subgroups 1 ~ N a G, including the case N = G. Now fix a minimal normal subgroup 1 ~ L a G; we know from Theorem 3.1 that L is not solvable, as X ~ _ 0. Hence L is a direct product of isomorphic non-abelian simple groups (see [8], Thm 2.1.5). Assume that L is not simple; by Lemma 3.4, for some simple factor H a L. Also, note that L = G) since it is a minimal normal subgroup. Now we have that for all g E G, hence applying the same lemma once again, but now to the L-action on X, we infer that X L ~ ~, a contradiction. So L is simple; now set H = Cc(L). Then we have that H a G (this follows from the fact that L a G) and so XH =1= 0, by Lemma 3.4. However we have assumed that the action is essential, whence H 1. D = The condition Cc(L) = 1 is equivalent to G C Aut(L). Using this proposi- tion we can decide which groups admit essential fixed point free actions on acyclic 2-dimensional complexes by first determining the simple groups with such actions and then looking at automorphism groups only for that restricted collection. As the proof of the main theorem will require explicit knowledge about the finite simple groups, it seems appropriate to briefly recall their classification, we refer to [9] for a detailed explanation. We should point out that it is by now common knowledge that complete details of the proof of the Classification Theorem were not available when it was announced in 1981; crucial work involving the so-called quasithin groups was never published and is known to contain gaps. Fortunately this has been resolved thanks to more recent work by Aschbacher and Smith and although a full account has not yet been published, a draft of their manuscript (over 1200 pages long!) is now available on the world wide web (see ~3~ ) . The following theorem encapsulates our understanding of finite simple groups, and its proof requires literally thousands of pages of mathematical arguments by many authors. THEOREM 3.7. - Let L denote a non-abelian finite sample group, then it must be isomorphic to one of the following groups: - an alternating group An for n > 5 - a finite group of Lie type, i.e. a finite Chevalley group or a twisted analogue(4) - one of the 26 sporadic simple groups. 4. TECHNIQUES FOR CONSTRUCTING ACTIONS One of the main results in [15] is an explicit listing of conditions which imply the existence of fixed point free actions on acyclic complexes. We first introduce (4)We should mention that the Tits group 2F4 (2)’ is actually of index 2 in the full Lie type group 2F4(2). DEFINITION 4.1. - A non-empty f amily~5~ ,~ of subgroups of a group G is said to be separating if it has the following three properties: (a) G ~ 0; (b ) any subgroup of an element in F is in 0; and (c) for any H a K C G with KI H solvable, K ~ F if It is not hard to see that any maximal subgroup in a separating family of subgroups of G is self-normalizing. If G is solvable, then it has no separating family of subgroups. For G not solvable we let S,CV denote the family of solvable subgroups, which is the minimal separating family for G. DEFINITION 4.2. - Given G and a f amily of subgroups 0, a is a G-complex such that all of its isotropy subgroups lie in 0. It is said to be universal (respectively H-universal) if the fixed point set of each is contractible (respec- tively acyclic). The following proposition relates the two previous concepts in our situation. PROPOSITION 4.3. - Let X denote a 2-dimensional acyclic G-complex without fixed points. Let F _ ~ H C G X H ~ ~ ~ . Then 0 is a separating f amily of subgroups of G, and X is an H-universal (G, Given a family of subgroups 0, let N(0) denote the nerve of F (regarded as a poset via inclusion) with a G-action induced by conjugation. Given any set H of subgroups in G, we let denote the poset of those subgroups in F which contain some element of H. For a single subgroup H we use the notation and to denote the posets of subgroups containing H or strictly containing H, respectively. We denote X x = The following are two key technical lemmas which will be required: LEMMA 4.4. - If X denotes a universal (H-universal) (G, .~’)-complex then there exists a G-map X N(0) which induces a homotopy equivalence (homology equiva- lence) between X ~ and LEMMA 4.5. - Let F be any f amily of subgroups of G, and let .~o C .~’ be any sub f amily such that N(,~’~H ) N ~ for all H E 0 - 00. Then any (H-)universal (G, F0)-complex is also an (H-)universal (G, and N((F0)H) ~ ’ ’ for any set H of subgroups of G. A complex Y is said to be homologically m-dimensional if Hn(X, Z) 0 for all = n > m and Hm(X, Z) is 7 -free. For later use we observe that, for 1, if X is an m-dimensional acyclic complex, then any subcomplex of X is homologically (m - 1 )-dimensional and that the intersection of a finite number of homologically (m - 2)-dimensional complexes is also homologically (m - 2)-dimensional. (5) A family is a collection of subgroups of a group G which is closed under conjugation. The following is a crucial criterion for the constructions we are seeking. PROPOSITION 4.6. - Let G be any finite group and let .~’ be a separating f amily for G. Then the following are equivalent: - There is a (finite) 2-dimensional H-universal (G, - is homologically 1-dimensional for each subgroup H E 0. - is homologically 1-dimensional for every set ~l of subgroups of G. Given a separating family ,~’ of subgroups of G, we say that H E .~ is a critical subgroup if is not contractible. Given the above, we can concentrate our attention on the family S,CV and its subfamily of critical subgroups, denoted First we record conditions which allow one to show that certain subgroups in a family are not critical. LEMMA 4.7. - Let ,I be any family if subgroups of G which has the property that H C H’ C H" and H, H" E .~ imply that H’ E 0. Fix a subgroup H E 0; then is contractible if any of the following holds: - H is not an intersection of maximal subgroups in 0. - There is a subgroup H E .~’ properly containing H and such that H C K n H for We can now state a simple sufficient condition for the existence of a 2-dimensional H-universal (G, 0)-complex: PROPOSITION 4.8. - Let .~ be any separating family of subgroups of G. Assume for every non-maximal critical subgroup 1 ~ H E 0, that Nc(H) E 0, and that H C for all non-maximal critical subgroups properly containing H. Then there exists a 2-dimensiorcal H-universal (G, F)-complex. We can in fact give a concrete description of the complex. For this we must introduce an integer associated to H E 0. DEFINITION 4.9. - If H E 0, a f amily of subgroups of G, we define Now let be conjugacy classes representatives for the maximal sub- groups of 0, and let Hi , ... , Hk be conjugacy class representatives for all non-maximal critical subgroups of 0. Then there is a 2-dimensional H-universal (G, 0)-universal complex X which consists of one orbit of vertices of type G/Mi for each 1 i n, of 1-cells of type G/Hj for each 1 j k, and free orbits of 1- and 2-cells. If G is simple(6) then X can be constructed to contain exactly free orbits of 2-cells, and no free orbits of 1-cells. (6) In fact G must satisfy an additional technical condition which does not affect the results here. 5. EXPLICIT ACTIONS In this section we will outline the construction of fixed point free actions on acyclic 2-dimensional complexes for the simple groups PSL2(2k), for k 2; PSL2 (q) for q - ~3 (mod 8) and ? ~ 5; and for Sz(2~) for odd l~ > 3. Example 5.1. - Let G = PSL2 (q), where q = 2k and k 2. Then there is a 2-dimensional acyclic fixed point free G-complex X all of whose isotropy subgroups are solvable. The complex X can be constructed with three orbits of vertices, with isotropy subgroups isomorphic to B Fq x Cq-l, D2(g-i) and D2(q+1); three orbits = of edges with isotropy subgroups isomorphic to Cq-l, C2 and C2 ; and one free orbit of 2-cells. Here B denotes a Borel subgroup, expressed as a semi-direct product isomorphic to (C2)k ~a Cq-l, identified with the subgroup of projectivized upper triangular matrices. In our notation Cr denotes the cyclic group of order r and Dr denotes the dihedral group of order r. In fact can be identified with the subgroup of monomial matrices. This example can be explained from the following analysis. The conjugacy classes of maximal solvable subgroups of G are represented precisely by the groups B, D2(g-i)> and D2(q+1) . The non-maximal critical subgroups must be intersections of maximal subgroups, one can check that up to conjugacy we get Cq-1 , C2 and 1. The precise numbers of orbits which appear is determined by calculating the integers is£v(H) for the isotropy subgroups. This example can actually be constructed directly using the 1-skeleton Y1 of the coset complex Y for the triple of subgroups (Kl, K2, K3) _ (B, D2(q_1), D2(q+1)) in G = PSL2(Fg) given by the maximal solvable subgroups. We can describe Y as the G-complex with vertex set G/ K1UG/ K2UG/ K3, where G acts by left translation, and with a 1-simplex for every pair of cosets with non-empty intersection and a 2-simplex for every triple of cosets with non-empty intersection. The following picture describes the orbit space Y/G: It is not hard to see that as G = (Kl, K2, K3 ) , the complex Y is connected; however (as shown in [2], § 9) it is not acyclic for 1~ > 3, where q = 2~. However, one can show that the module Hl (Yl, Z) is stably free - this involves a geometric argument based on the fact that Y1 is a graph such that the fixed point sets are either contractible or empty for all subgroups 1 ~ H C G and contractible for all p-subgroups in G. The fact that G is a nonabelian simple group implies that the module must in fact be free (for a proof see [15], Prop. C.4.). Now we can simply attach a single free G-cell to Y1 to kill its homology, yielding the acyclic complex X. Carrying out this analysis in the classical case G = A5 yields an acyclic complex X (which in this case is actually identical to the complex Y) whose cellular chains give a complex of the form(7) Example 5.2. - Let G = PSL2(Fq), where q = 5 and q == ~3 (mod 8). Then there exists a 2-dimensional acyclic fixed point free G-complex X, all of whose isotropy subgroups are solvable. More precisely, X can be constructed to have four orbits of vertices with isotropy subgroups isomorphic to Fq x C~q_1~~2, Dq-l, Dq+i and A4; four orbits of edges with isotropy subgroups isomorphic to C~q_1)~2, C2 x C2, C3 and C2; and one free orbit of 2-cells. These examples are slightly more complicated as the structure of the complex will depend on the value of q modulo 8. Before explaining the final set of examples, we briefly recall the structure of the Suzuki groups Sz(q) (see [6], ~11~, [19] for details). Fix q = 22’~+1 and let 9 E Aut(Fq) be the automorphism x8 = X2k+l = x~ (note that (x8)8 = X2). For a, b E Fq and A E (I~q ) * , define the elements (7) If we consider the original construction discussed in § 1 of an acyclic A5-complex, then one can obtain the cellular structure below by subdividing each pentagon into a union of ten triangles. ASTERISQUE 290