# Surface group amalgams that (don't) act on 3-manifolds

## TL;DR

This paper classifies certain surface group amalgams based on their ability to act as 3-manifold groups, showing they are virtually 3-manifold groups and relating them to right-angled Coxeter groups.

## Contribution

It identifies which surface group amalgams are not 3-manifold groups and proves their virtual 3-manifold group status, also establishing their commensurability with right-angled Coxeter groups.

## Key findings

- Certain surface amalgams are not 3-manifold groups.
- All considered amalgams are virtually 3-manifold groups.
- Surface amalgams are commensurable with right-angled Coxeter groups.

## Abstract

We determine which amalgamated products of surface groups identified over multiples of simple closed curves are not fundamental groups of 3-manifolds. We prove each surface amalgam considered is virtually the fundamental group of a 3-manifold. We prove that each such surface group amalgam is abstractly commensurable to a right-angled Coxeter group from a related family. In an appendix, we determine the quasi-isometry classes among these surface amalgams and their related right-angled Coxeter groups.