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3-Manifold Groups Are Virtually Residually \(p\)
Matthias Aschenbrenner, University of California, Los Angeles, CA, and Stefan Friedl, University of Koln, Germany

Memoirs of the American Mathematical Society
2013; 100 pp; softcover
Volume: 225
ISBN-10: 0-8218-8801-3
ISBN-13: 978-0-8218-8801-8
List Price: US$69
Individual Members: US$41.40
Institutional Members: US$55.20
Order Code: MEMO/225/1058
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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.

Table of Contents

  • Introduction
  • Preliminaries
  • Embedding theorems for \(p\)-Groups
  • Residual properties of graphs of groups
  • Proof of the main results
  • The case of graph manifolds
  • Bibliography
  • Index
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