Dave Morris

(University of Lethbridge)

Quasi-Isometric Bounded Generation

Date Friday, March 1, 2019

A subset \(X\) "boundedly generates" a group \(G\) if every element of \(G\) is the product of a bounded number of elements of \(X\). This is a very powerful notion in abstract group theory, but geometric group theorists (and others) may also need a good bound on the sizes of the elements of \(X\) that are used. (We do not want to have to use large elements of \(X\) to represent a small element of \(G\).) Twenty-five years ago, Lubotzky, Mozes, and Raghunathan proved an excellent result of this type for the case where \(G\) is the group \(SL(n,\mathbb{Z})\) of \(n\times n\) matrices with integer entries and determinant one, and \(X\) consists of the elements of the natural copies of \(SL(2,\mathbb{Z})\) in \(G\). We will explain the proof of this result, and discuss a recent generalization to other arithmetic groups.