Topology Seminar: Emily Hamilton (Cal Poly SLO), 'Separability of Double Cosets and Conjugacy Classes in 3-Manifold Groups'

Event Date: 

Tuesday, April 8, 2014 - 3:30pm to 4:30pm

Event Location: 

  • 4607B South Hall

Abstract: A subset $X$ of a group $Gamma$ is {it separable} in $Gamma$ if for every element $gamma in Gamma - X$ there is a homomorphism $phi$ from $Gamma$ to a finite group such that $phi(gamma) notin phi(X)$. A group $Gamma$ is {it residually finite} if the trivial subgroup is separable, {it subgroup separable} if every finitely generated subgroup of $Gamma$ is separable, and {it conjugacy separable} if every conjugacy class in $Gamma$ is separable. Separability has applications in group theory and geometric topology. If a finitely presented group $Gamma$ is residually finite, then there exists an algorithm to decide if a given word in the presentation of $Gamma$ is trivial. If $G$ is subgroup separable, then one can solve more generalized word problems. In the context of geometric topology, subgroup separability has been used to solve immersion to embedding problems. For example, in $3$-manifold topology it is well known that subgroup separability allows passage from immersed incompressible surfaces to embedded incompressible surfaces in finite covers. In this talk we consider separability of double cosets and conjugacy classes in $3$-manifold groups. Let $M = {Bbb H}^3 / Gamma$ be a hyperbolic $3$-manifold of finite volume. We show that if $H$ and $K$ are abelian subgroups of $Gamma$ and $g in Gamma$, then the double coset $HgK$ is separable in $Gamma$. As a consequence, we prove that if M is a closed, orientable Haken $3$-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 recent work of Agol and Wise, it follows that if $M$ is a compact, orientable $3$-manifold, then $pi_1(M)$ is conjugacy separable.