相关论文: Expansion in finite simple groups of Lie type

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^(1+epsilon) where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $ A $ of a finite simple group $ G $ of Lie type of…

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a…

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense.…

We prove that there exist $k\in N$ and $0<\epsilon\in R$ 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 $\epsilon$-expander.

Let G be a finitely presented group, and let {G_i} be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1. G_i is an amalgamated free product or…

We prove that all isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work…

We study the representations of non-commutative universal lattices and use them to compute lower bounds for the \TauC for the commutative universal lattices $G_{d,k}= \SL_d(\Z[x_1,...,x_k])$ with respect to several generating sets. As an…

Finite simple groups of Lie type as expanders

We prove that if G is SL_2(F) or PSL_2(F), where F is a finite field, and A is a set of generators of G, then either |AAA| > |A|^(1+epsilon), where epsilon is an absolute positive real number, or AAA=G. As a corollary we get that the…

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a…

We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set. We answer a question of Pyber and Szab\'o (2013) by exhibiting a sequence of finite groups $G_n$ with $|G_n| \to \infty$ admitting…

We construct an explicit generating sets $F_n$ and $\tilde F_n$ of the alternating and the symmetric groups, which make the Cayley graphs $C(Alt(n), F_n)$ and $C(Sym(n), \tilde F_n)$ a family of bounded degree expanders for all sufficiently…

In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an N-series of subgroups. The asymptotics of the Poincare series of this algebra give estimates on the growth of the group…

We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This…

We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.

We construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of $SL_2 (\mathbb{F}_p [t])$ modulo certain square-free ideals. We describe some applications of our results…

We produce new short laws in two variables valid in finite groups of Lie type. Our result improves upon results of Kozma and the second named author, and is sharp up to logarithmic factors, for all families except possibly the Suzuki…

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness…