Separability of double cosets and conjugacy classes in 3-manifold groups

Details Separability of double cosets and conjugacy classes in 3-manifold groups Separability of double cosets and conjugacy classes in 3-manifold groups

2011-09-13

arxiv.org/abs/1109.2765v1

Mathematics

Group Theory

25 pages

Scientific paper

Let M = H^3/\Gamma be a hyperbolic 3-manifold of finite volume. We show that if H and K are abelian subgroups of \Gamma and g is an element of \Gamma, then the double coset HgK is separable in \Gamma. As a consequence we prove that if M is a closed 3-manifold and the fundamental group of every hyperbolic piece of the torus decomposition of M is conjugacy separable then so is the fundamental group of M. Invoking an announcement of D. Wise, it follows that if M is a closed, irreducible, Haken 3-manifold then either \pi_1(M) is conjugacy separable or M is a closed hyperbolic 3-manifold and the virtual first Betti number of M is less than or equal to one.

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