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$3$-Manifold Groups are Virtually Residually $p$

About this Title

Matthias Aschenbrenner, University of California, Los Angeles, California, USA and Stefan Friedl, Mathematisches Institut, Universität zu Köln, Germany

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 225, Number 1058
ISBNs: 978-0-8218-8801-8 (print); 978-1-4704-1058-2 (online)
Published electronically: February 19, 2013
Keywords: $3$-manifolds, fundamental groups, virtually residually $p$
MSC: Primary 20E26, 57M05; Secondary 20E06, 20E22

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Table of Contents


  • Introduction
  • 1. Preliminaries
  • 2. Embedding Theorems for $p$-Groups
  • 3. Residual Properties of Graphs of Groups
  • 4. Proof of the Main Results
  • 5. The Case of Graph Manifolds


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.

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