3-manifold Groups Are Virtually Residually P
Author： Matthias Aschenbrenner;Stefan Friedl
Publisher： American Mathematical Society
Publication year： 2013
E-ISBN: 9781470410582
P-ISBN(Paperback): 9780821888018
P-ISBN(Hardback): 9780821888018
Subject： O152 group theory
Keyword： Algebra and Algebraic Geometry
Language： ENG
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3-manifold Groups Are Virtually Residually P
Description
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 generalisation 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.