Henry Wilton

3-Manifold Groups

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

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

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 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.

Some virtually special hyperbolic 3-manifold groups

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem of Haglund and Wise to the relatively hyperbolic setting and deduce that the fundamental group of M is LERF and that the geometrically finite subgroups of the fundamental group are virtual retracts. Examples of 3-manifolds admitting such a decomposition include augmented link complements. We classify the low-complexity augmented links and describe an infinite family with complements not commensurable to any 3-dimensional reflection orbifold.

On surface subgroups of doubles of free groups

We give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.

Enumerating limit groups

We prove that the set of limit groups is recursively enumerable, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (a la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmically calculate centralizers in relatively hyperbolic groups. Applications include the existence of recognition algorithms for limit groups and free groups.

Conjugacy classes of solutions to equations and inequations over hyperbolic groups

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsion-free hyperbolic group.

Geometric and Cohomological Methods in Group Theory: Solutions to Bestvina & Feighn's exercises on limit groups

This article gives solutions to the exercises in Bestvina and Feighns paper on Selas work on limit groups. We prove that all constructible limit groups are limit groups and give an account of the shortening argument of Rips and Sela.

Limit groups, positive-genus towers and measure equivalence

By definition, an

POLYGONAL WORDS IN FREE GROUPS

\omega

One-ended subgroups of graphs of free groups with cyclic edge groups

-residually free tower is positive-genus if all surfaces used in its construction are of positive-genus. We prove that every limit group is virtually a subgroup of a positive-genus,