Quantifying Residual Finiteness of Linear Groups

TL;DR

This paper investigates how well linear groups can be approximated by finite groups, providing bounds on residual finiteness growth that depend on the group's linear degree and specific cases with precise asymptotics.

Contribution

It establishes an upper bound on residual finiteness growth for linear groups based on their degree, and computes exact growth rates for subgroups of higher rank Chevalley groups.

Findings

Residual finiteness growth for linear groups is bounded by (n log n)^{d^2-1}.

Finite index subgroups of G(Z) and G(F_p[t]) have growth rate n^{dim(G)}.

Precise asymptotics are provided for subgroups of higher rank Chevalley groups.

Abstract

Normal residual finiteness growth measures how well a finitely generated group is approximated by its finite quotients. We show that any linear group $\Gamma \leq \mathrm{GL}_d(K)$ has normal residual finiteness growth asymptotically bounded above by $(n\log n)^{d^2-1}$; notably this bound depends only on the degree of linearity of $\Gamma$. We also give precise asymptotics in the case that $\Gamma$ is a subgroup of a higher rank Chevalley group $G$ and compute the non-normal residual finiteness growth in these cases. In particular, finite index subgroups of $G(\mathbb{Z})$ and $G(\mathbb{F}_p[t])$ have normal residual finiteness growth $n^{\dim(G)}.$