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 ≤ 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 Γ. If char K = 0 or K is a purely transcendental extension of a finite field, then this bound can be improved to n d 2 −1 . We also give lower bounds on the normal residual finiteness growth of Γ in the case that Γ 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(F p [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.

Effective subgroup separability of finitely generated nilpotent groups

“…This result was later extended in the context of arithmetic groups defined over purely transcendental extensions of a finite field by Franz [22].…”

“…The following theorem by Franz is the best upper bound which has be provided for general affine algebraic group schemes defined over Z. The upper bound for fields of characteristic 0 was provided by Bou-Rabee and Kaletha [8] and was later extended to fields of characteristic p by Franz [22]. While we mention this theorem in the context of Chevalley groups of arbitrary rank, the theorem applies to broader class of all affine algebraic group schemes defined over Z.…”