3-manifold groups are virtually residually p

TL;DR

This paper proves that all 3-manifold groups are virtually residually p for all but finitely many primes p, extending known results and supporting the conjecture that these groups are linear.

Contribution

It establishes that every 3-manifold group is virtually residually p for all but finitely many primes p, generalizing previous special cases.

Findings

Fundamental groups of hyperbolic 3-manifolds are virtually residually p.

All 3-manifold groups are virtually residually p for all but finitely many p.

Supports Thurston's conjecture on linearity of 3-manifold groups.

Abstract

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 we 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.