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.