# Profinite non-rigidity of arithmetic groups

## TL;DR

This paper demonstrates that for many high rank arithmetic lattices, there are finite index subgroups with isomorphic profinite completions but not isomorphic themselves, revealing non-rigidity phenomena.

## Contribution

It establishes the existence of non-rigid behaviors in the profinite completions of high rank arithmetic groups, with specific exceptions identified.

## Key findings

- Existence of non-isomorphic subgroups with isomorphic profinite completions
- Non-rigidity phenomenon in high rank arithmetic lattices
- Identification of exceptions to the non-rigidity rule

## Abstract

We show that for a typical high rank arithmetic lattice $\Gamma$, there exist finite index subgroups $\Gamma_{1}$ and $\Gamma_{2}$ such that $\Gamma_{1} \not\simeq \Gamma_{2}$ while $\widehat{\Gamma_{1}} \simeq \widehat{\Gamma_{2}}$. But there are exceptions to that rule.