Finite Simple Groups as Expanders

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.Comment: 10 page

Linear Approximate Groups

This is an informal announcement of results to be described and proved in detail in a paper to appear. We give various results on the structure of approximate subgroups in linear groups such as \SL_n(k). For example, generalising a result of Helfgott (who handled the cases $n = 2$ and 3), we show that any approximate subgroup of \SL_n(\F_q) which generates the group must be either very small or else nearly all of \SL_n(\F_q). The argument is valid for all Chevalley groups G(\F_q).Comment: 11 pages. Submitted, Electronic Research Announcements. Small change