A faithful representation and profinite completion

Abstract

We exhibit two finitely generated residually finite groups $G_1$ and $G_2$ with isomorphic profinite completions $\widehat{G_1} \cong \widehat{G_2}$, such that $G_1$ has a faithful representation into $GL(d,\mathbb{C})$ but $G_2$ is not linear. This answers a question of Andrei Jaikin-Zapirain, Alexander Lubotzky, and Martin Bridson negatively. We also present two finitely generated residually finite groups $H_1$ and $H_2$ with isomorphic profinite completions $\widehat{H_1} \cong \widehat{H_2}$ satisfying the following. For every field $k$ with characteristic $p \geq 5$ and transcendence degree $r \geq 2$, there exists $d>0$ such that $H_1$ has a faithful representation into $GL(d,k)$. However, $H_2$ admits no faithful finite-dimensional representation over any field of positive characteristic.