Subgroup separability of amalgamated products of hyperbolic 3-manifolds
by
Emily Hamilton
Emory University

Let H be a subgroup of a group G. H is separable in G if given any element g in G \H, there is a finite index subgroup K subset G, such that H subset K but g not in K. A group G is subgroup separable if every finitely generated subgroup of G is separable in G. Subgroup separability is a powerful property. It has applications in group theory and geometric topology. In group theory, it is linked to the solution of generalized word problems. In geometric topology, it is the traditional group-theoretic tool used to decide if a given immersion in a manifold M will lift to an embedding in a finite covering of M.

In this talk we examine separability of amalgamated products of hyperbolic 3-manifolds. In particular, we consider the following open question. Let M₁ and M₂ be compact orientable 3-manifolds with non-empty boundary whose interiors admit complete hyperbolic structures of finite volume. Fix boundary components T₁ in M₁ and T₂ in M₂, and let f:T₁ --> T₂ be a homeomorphism. Let M be the manifold obtained by identifying M₁ and M₂ along the fixed boundary components via f. If A is a separable subgroup of \pi₁(M₁) and B is a separable subgroup of \pi₁(M₂), is <A, B > a separable subgroup of \pi₁(M)? We give an affirmative answer in the case where A and B are cyclic subgroups generated by loxodromic elements.

Date received: January 31, 2001