Quantifying residual finiteness of arithmetic groups

TL;DR

This paper investigates the residual finiteness of arithmetic groups by analyzing their normal Farb growth, establishing a precise growth rate for S-arithmetic subgroups of higher rank Chevalley groups.

Contribution

It proves that S-arithmetic subgroups of higher rank Chevalley groups have a normal Farb growth proportional to n raised to the dimension of the group, providing a quantitative measure.

Findings

Normal Farb growth of these groups is n^dim(G).

Provides a quantitative characterization of residual finiteness.

Enhances understanding of finite quotient approximations of arithmetic groups.

Abstract

The normal Farb growth of a group quantifies how well-approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal Farb growth n^dim(G).