# 3-Manifold Groups Are Virtually Residually \(p\)

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*Matthias Aschenbrenner; Stefan Friedl*

Given a prime \(p\), a group is called residually \(p\) if the intersection of its \(p\)-power index normal subgroups is trivial. A group is called virtually residually \(p\) if it has a finite index subgroup which is residually \(p\). It is well-known that finitely generated linear groups over fields of characteristic zero are virtually residually \(p\) for all but finitely many \(p\). In particular, fundamental groups of hyperbolic \(3\)-manifolds are virtually residually \(p\). It is also well-known that fundamental groups of \(3\)-manifolds are residually finite. In this paper the authors prove a common generalization of these results: every \(3\)-manifold group is virtually residually \(p\) for all but finitely many \(p\). This gives evidence for the conjecture (Thurston) that fundamental groups of \(3\)-manifolds are linear groups.