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 $(n \times n)$-matrix with $n \geq 3$ is a product of a bounded number of elementaries (Carter-Keller) which yields (BG) for $SL_n(\mathbb{Z})$ $(n \geq 3)$. Next, we will present a geometric method for proving (BG) for $S$-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 $SL_n(\mathbb{C}[x])$ is not a bounded product of elementaries.