Groups with bounded generation: properties and examples

IAS seminar on Arithmetic Groups

After surveying some important consequences of the property of bounded generation (BG) dealing with SS-rigidity, the congruence subgroup problem, etc., we will focus on examples of boundedly generated groups. We will prove that every unimodular (n×n)-matrix with n≥3 is a product of a bounded number of elementaries (Carter-Keller) which yields (BG) for SLn(Z) (n≥3). Next, we will present a geometric method for proving (BG) for SS-arithmetic subgroups of orthogonal groups (Erovenko-R.) which applies also to some other groups of classical types. Time permitting, we will also discuss the non-example, due to W. van der Kallen, that SLn(C[x]) is not a bounded product of elementaries.