Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5771980 | Journal of Algebra | 2017 | 37 Pages |
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 Îâ¤GLd(K) has normal residual finiteness growth asymptotically bounded above by (nlogâ¡n)d2â1; notably this bound depends only on the degree of linearity of Î. If char K=0 or K is a purely transcendental extension of a finite field, then this bound can be improved to nd2â1. We also give lower bounds on the normal residual finiteness growth of Î in the case that Î 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 ndimâ¡(G). We also compute the non-normal residual finiteness growth in the above cases; for the lower bounds the exponent dimâ¡(G) is replaced by the minimal codimension of a maximal parabolic subgroup of G.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Daniel Franz,