Profinite rigidity in lattices of $\mathrm{PSL}(2,\mathbb{C})$

Abstract

A finitely generated group is profinitely rigid among a class of finitely generated groups if it can be distinguished among this class by its set of finite quotient groups. This paper proves that all lattices in $\mathrm{PSL}(2,\mathbb{C})$ are profinitely rigid among themselves. In addition, for any lattice $\Gamma\le \mathrm{PSL}(2,\mathbb{C})$, it is proven that $\mathrm{Out}(\widehat{\Gamma})\cong \mathrm{Out}(\Gamma)$, where $\widehat{\Gamma}$ denotes the profinite completion of $\Gamma$.