Quantifying Residual Finiteness of Linear Groups

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 $n^{\dim(G)}.$