Arithmetic groups with isomorphic finite quotients

TL;DR

This paper investigates the relationship between S-arithmetic groups and their finite quotients, showing that many such groups share the same finite quotients and that the classification map has fibers of unbounded size.

Contribution

It demonstrates that for a broad class of S-arithmetic groups, the map to their profinite completions is finite-to-one with unbounded fiber sizes, revealing new structural insights.

Findings

Finite-to-one correspondence for many S-arithmetic groups

Existence of unbounded fiber sizes in the classification map

Profinite completions characterize finite quotients of these groups

Abstract

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the the fibers are of unbounded size.