Quantifying residual finiteness of linear groups

Abstract

Normal residual finiteness growth measures how well a finitely generated residually finite group is approximated by its finite quotients. We show that any finitely generated linear group Γ≤GLd(K) has normal residual finiteness growth asymptotically bounded above by (nlog⁡n)d2−1; notably this bound depends only on the degree of linearity of Γ. If char K=0 or K is a purely transcendental extension of a finite field, then this bound can be improved to nd2−1. We also give lower bounds on the normal residual finiteness growth of Γ in the case that Γ is a finite index subgroup of G(Z) or G(Fp[t]), where G is Chevalley group 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 ndim⁡(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.