Quantifying the Residual Finiteness of Linear Groups

A residually finite group is a group for which the intersection of all finite index subgroups is trivial; such a group can be studied using its finite quotients. Normal residual finiteness growth measures how well a finitely generated residually finite group is approximated by its finite quotients. We show that any linear group $\Gamma\leq \GL_d(K)$ 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 $\Gamma$. If char $K=0$, then this bound can be improved to $n^{d^2-1}$. We also give lower bounds on the normal residual finiteness growth of $\Gamma$ in the case that $\Gamma$ is a finitely generated subgroup of a Chevalley group $G$ of rank at least 2. These lower bounds agree with the computed upper bounds, providing exact asymptotics on the normal residual finiteness growth. In particular, finite index subgroups of $G(\Z)$ and $G(\Fp[t])$ have normal residual finiteness growth $n^{\dim(G)}.$ We also compute the non-normal residual finiteness growth in the above cases; for the lower bounds the exponent $\dim(G)$ is replaced by the minimal codimension of a maximal parabolic subgroup of $G$.