There is a constant C 0 C_0 such that all nonabelian finite simple groups of rank n n over F q \mathbb {F}_q , with the possible exception of the Ree groups 2 G 2 ( 3 2 e + 1 ) ^2G_2(3^{2e+1}) , have presentations with at most C 0 C_0 generators and relations and total length at most C 0 ( log ⁡ n + log ⁡ q ) C_0(\log n +\log q) . As a corollary, we deduce a conjecture of Holt: there is a constant C C such that dim ⁡ H 2 ( G , M ) ≤ C dim ⁡ M \dim H^2(G,M) \leq C\dim M for every finite simple group G G , every prime p p and every irreducible F p G {\mathbb F}_p G -module M M .

Abstract. All finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2 G 2 (q), have presentations with at most 49 relations and bit-length O(log n + log q). Moreover, A n and S n have presentations with 3 generators, 7 relations and bitlength O(log n), while SL(n, q) has a presentation with 6 generators, 25 relations and bit-length O(log n + log q).

Abstract. We prove the following three closely related results:(1) Every finite simple group G has a profinite presentation with 2 generators and at most 18 relations. (2) If G is a finite simple group, F a field and M an F G-module, then dim

Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F . We give a solution to the latter question using elementary techniques which rely purely on the description of F as the group of piecewise linear orientationpreserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.2010 Mathematics Subject Classification. primary 20F10; secondary 20E45, 37E05.

We study the representations of non-commutative universal lattices and use them to compute lower bounds for the τ -constant for the commutative universal latticeswith respect to several generating sets.As an application of the above result we show that the Cayley graphs of the finite groups SL 3k (Fp) can be made expanders using suitable choice of the generators. This provides the first examples of expander families of groups of Lie type where the rank is not bounded and gives a natural (and explicit) counter examples to two conjectures of Alex Lubotzky and Benjamin Weiss.

We construct explicit generating sets Sn andSn of the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), Sn) and C(Sym(n),Sn) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling and other areas.

We prove that there exist k ʦ ‫ގ‬ and 0 < ʦ ‫ޒ‬ such that every non-abelian finite simple group G, which is not a Suzuki group, has a set of k generators for which the Cayley graph Cay(G; S) is an -expander.expander graphs ͉ Ramanujan complexes L et X be a finite graph and 0 Ͻ ʦ ‫.ޒ‬ Then X is called an -expander if for every subset A of the vertices of X with ͉A͉ Յ (1͞2)͉X͉ we have ͉ѨA͉ Ն ͉A͉, where ѨA denotes the boundary of A, i.e., the vertices of distance 1 from A.Expander graphs play an important role in computer science and combinatorics, and many efforts has been dedicated to their constructions (cf. ref. 1, the references therein, and ref.2). Many of these constructions are of Cayley graphs, and in particular various infinite families of finite simple groups have been shown to be expanding families.A finite group G is called an -expander with respect to a generating subset S if the Cayley graph Cay(G; S) is an -expander. We will say that an infinite family G of groups is a family of expanders if there exists k ʦ ‫ގ‬ and 0 Ͻ ʦ ‫ޒ‬ such that every group G ʦ G has a subset S of k generators with respect to (w.r.t.) which G is an -expander. In this situation we also say that the groups G ʦ G are expanders ''uniformly.'' Until recently all known such families consisting of simple groups were of bounded Lie rank (cf. refs. 1 and 3), a fact that has raised speculation that this is the only possibility (see refs. 4 and 5).It is easy to see that the diameter of each graph X i from an expander family {X i } is most c log͉X i ͉ (where c is a constant). In ref. 6 it was shown that every non-abelian finite simple group has a set S of seven generators for which the Cayley graph Cay(G; S) has a diameter bounded by C log(͉G͉) for an absolute constant C. It was conjectured there that one can even make all finite simple groups expanders uniformly, although, as observed by Y. Luz (see ref. 4), the generators used in ref. 6 do not give rise to expanders.The main goal of this note is to announce a proof of almost the whole of this conjecture. Theorem 1. There exist k ʦ ‫ގ‬ and 0 Ͻ ʦ ‫ޒ‬ such that every non-abelian finite simple group G, which is not a Suzuki group, has a set S of k generators for which Cay(G; S) is an -expander.In fact careful estimates using variations of some of the arguments below yield k Ͻ 1,000 and Ͼ 10 Ϫ10 .We believe that the above theorem holds also for the Suzuki groups, but our (diverse) methods do not apply to them. What makes them exceptional is the fact that they do not contain copies of SL 2 ‫ކ(‬ p ) or PSL 2 ‫ކ(‬ p ) like all the other finite simple groups (see below for more details).The proof of Theorem 1 is the accumulation of the works (refs. 7-10 and A.L., unpublished results) in the following chronological order.In ref. . This includes, in particular, the groups EL 3 (Mat n ‫ކ(‬ q )) Ӎ SL 3n ‫ކ(‬ q ), and thus SL 3n ‫ކ(‬ q ) are uniformly expanders for all n and for all prime powers q. For every d Ն 3, the group SL d ‫ކ(‬ q ) is a bounded product of copies of SL 3n ‫ކ(‬ q ) for n ϭ d͞3...

The observation that a graph of rank n can be assembled from graphs of smaller rank k with s leaves by pairing the leaves together leads to a process for assembling homology classes for Out(Fn) and Aut(Fn) from classes for groups Γ k,s , where the Γ k,s generalize Out(F k ) = Γ k,0 and Aut(F k ) = Γ k,1 . The symmetric group Ss acts on H * (Γ k,s ) by permuting leaves, and for trivial rational coefficients we compute the Ss-module structure on H * (Γ k,s ) completely for k ≤ 2. Assembling these classes then produces all the known nontrivial rational homology classes for Aut(Fn) and Out(Fn) with the possible exception of classes for n = 7 recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of Aut(Fn) and Out(Fn) which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.