Doubles of groups and hyperbolic LERF $3$-manifolds

Rita Gitik

Review from Zentralblatt MATH:

Let $G$ be a group. The profinite topology on $G$ is obtained by setting all finite index subgroups of $G$ to be the base of open neighborhoods of the identity element. $G$ is said to be LERF (locally extended residually finite) if any finitely generated subgroup of $G$ is closed in the profinite topology. A 3-manifold is said to be LERF if its fundamental group is LERF.

The main result of this paper is the following Theorem. Let $M$ be a compact hyperbolic LERF 3-manifold with boundary, which does not have boundary tori, let $B$ be a connected submanifold of $

tial M$, such that $B$ is incompressible in $M$, and let $D(M)$ be the double of $M$ along $B$. If $D(M)$ is hyperbolic, has nonempty boundary, and has no boundary tori, then $D(M)$ is LERF. If the boundary of $D(M)$ is empty, then any geometrically finite subgroup and any freely indecomposable geometrically infinite subgroup (hence any closed surface subgroup) of the fundamental group of $D(M)$ is closed in the profinite topology.

Using this result, the author constructs a family of hyperbolic 3-manifolds with LERF fundamental group.

Reviewed by A.Papadopoulos

Keywords: hyperbolic manifolds; profinite topology; hyperbolic groups; doubles of groups; LERF groups; LERF manifolds; subgroups of finite index; locally extended residually finite groups; finitely generated subgroups; fundamental groups

Classification (MSC2000): 20F67 20E07 20E26 57M07

Full text of the article:

• PDF file (261 kilobytes)