Quantifying residual finiteness of arithmetic groups. (Q2894208)

Let \(\Gamma\) be a finitely generated residually finite group. Given an element \(g\in\Gamma\), define its `depth' \(D_\Gamma(g)\) as the minimal order of a finite quotient of \(\Gamma\) in which \(g\) remains not equal to \(1\). The `normal residual finiteness growth function' \(F_{\Gamma,X}(n)\) is defined as the maximal depth of a non-identity element in \(\Gamma\) of length at most \(n\) with respect to a finite generating set \(X\). The asymptotic growth of this function is independent of \(X\). It is called the `normal residual finiteness growth' of \(\Gamma\). The normal residual finiteness growth quantifies how well approximated the group is by its finite quotients. Its study was initiated by the first author of this paper [in J. Algebra 323, No. 3, 729-737 (2010; Zbl 1222.20020)].NEWLINENEWLINE Let \(G\) be a Chevalley group of rank at least two. The main result of the present paper shows that the normal residual finiteness growth of every \(S\)-arithmetic subgroup of \(G\) is precisely \(n^{\dim(G)}\). This is somewhat surprising because in general the normal residual finiteness growth of a group and its finite index subgroup is not necessarily the same -- see Example~2.5 in the paper. The proofs use a wide variety of known results about algebraic groups and arithmetic subgroups.