Profinite non-rigidity of arithmetic groups

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.