Arithmetic groups
Quasi-isometric bounded generation by Q-rank-one subgroups
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,Z) of n-by-n matrices with integer entries and
determinant one, and X consists of the elements of the natural copies of SL(2,Z) in G.
We will explain the proof of this result, and discuss a recent generalization to other arithmetic groups.
