Random generation of finite and profinite groups and group enumeration

Abstract

The following article appeared in Annals of Mathematics 173.2 (2011): 769-814 and may be found at http://annals.math.princeton.edu/2011/173-2/p04We obtain a surprisingly explicit formula for the number of random elements needed to generate a finite d-generator group with high probability. As a corollary we prove that if G is a d-generated linear group of dimension n then cd + log n random generators suffice. Changing perspective we investigate profinite groups F which can be generated by a bounded number of elements with positive probability. In response to a question of Shalev we characterize such groups in terms of certain finite quotients with a transparent structure. As a consequence we settle several problems of Lucchini, Lubotzky, Mann and Segal. As a byproduct of our techniques we obtain that the number of r-relator groups of order n is at most ncr as conjectured by Mann.The first author is supported in part by the Spanish Ministry of Science and Innovation, the grant MTM2008-06680. The second author is supported in part by OTKA grant numbers NK 72523 and NK 78439