Profinite non-rigidity of arithmetic groups

We show that for a typical high rank arithmetic lattice $\mathrm{\Gamma }$, there exist finite index subgroups ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ such that ${\mathrm{\Gamma }}_1\simeq ̸{\mathrm{\Gamma }}_2$ while $\widehat{{\mathrm{\Gamma }}_1}\simeq \widehat{{\mathrm{\Gamma }}_2}$. But there are exceptions to that rule.