Short presentations of finite simple groups

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 response 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 surprizing 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 (to appear
in the J. of the AMS 2008. See arXiv:math.DG/0602508).