# Dave Morris

## 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.