The p-approximation property for simple Lie groups with finite center

Locally finite basic classical simple Lie superalgebras

In this work, we study direct limits of finite dimensional basic classical simple Lie superalgebras and obtain the conjugacy classes of Cartan subalgebras under the group of automorphisms.

Finite Groups with p-Sylow Coverings

Growth 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| where ε depends only on the Lie rank of L, or AAA = L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. Combining our result on growth with known results of Bourgain, Gamburd and Va...

Expansion in finite simple groups of Lie type

The beautiful book of Terry Tao starts with the following words: Expander graphs are a remarkable type of graph (or more precisely, a family of graphs) on finite sets of vertices that manage to simultaneously be both sparse (low-degree) and “highly connected” at the same time. They enjoy very strong mixing properties: if one starts at a fixed vertex of an (two-sided) expander graph and randomly...

Finite Simple Groups of Lie Type as Expanders

are uniform expanders. Nikolov [N] proved that every classical group is a bounded product of SLn(q)’s (with possible n = 2, but the proof shows that if the Lie rank is sufficiently high, say ≥ 14, one can use SLn(q) with n ≥ 3). Bounded product of expander groups are uniform expanders. Thus together, their results cover all classical groups of high rank. So, our Theorem is new for classical gro...