"Short presentations of finite simple groups"
Abstract:Finding 'nice & compact' presentations of various groups has been a subject of great interest for groups theorists for more than a century. Well known presentations are the Coxeter presentation of the finite symmetric groups and Steinberg presentation of groups of Lie type. In respond to conjectures of Babai and Szemeredi on one hand (motivated by questions in computational group theory) and of Mann on the other hand (motivated by questions on subgroup growth) we show that all non-abelian finite simple groups (with the possible exception of Ree groups) have presentations which are small (bounded number of relations|) and short (w.r.t the length of the relations). This is very suprising as the simple abelian groups- the cyclic groups of prime order- do not have such presentations!
We will describe the motivations and results, a cohomological application (proving a conjecture of Holt) and some connections with discrete subgroups of Lie groups and topology.
Joint work with Bob Guralnick, Bill Kantor and Martin Kassabov.