# Geometrically finite amalgamations of hyperbolic 3-manifold groups are not LERF

## TL;DR

This paper demonstrates that amalgamations of hyperbolic 3-manifold groups along geometrically finite subgroups are not LERF, extending previous results and implications to higher-dimensional hyperbolic manifolds.

## Contribution

It generalizes nonLERFness results for amalgamations of hyperbolic 3-manifold groups to include all geometrically finite subgroups and explores consequences for higher-dimensional hyperbolic manifolds.

## Key findings

- Amalgamations along geometrically finite subgroups are not LERF.
- Closed arithmetic hyperbolic 4-manifolds have nonLERF fundamental groups.
- Higher-dimensional hyperbolic manifolds (dimension ≥4) generally have nonLERF fundamental groups.

## Abstract

We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic $3$-manifold groups along abelian subgroups. A consequence of this result is that closed arithmetic hyperbolic $4$-manifolds have nonLERF fundamental groups. Along with the author's previous work, we get that, for any arithmetic hyperbolic manifold with dimension at least $4$, with possible exceptions in $7$-dimensional manifolds defined by the octonion, its fundamental group is not LERF.