Surface subgroups of graph products of groups

TL;DR

This paper investigates the structure of graph products of groups, showing that certain kernels are special and that specific graph products contain hyperbolic surface groups, with classifications for small graphs.

Contribution

It proves that graph product kernels are special and characterizes when graph products contain hyperbolic surface groups, especially for small defining graphs.

Findings

Graph product kernels are special.

Certain graph products contain hyperbolic surface groups.

Complete classification for small right-angled Coxeter groups.

Abstract

A graph product kernel means the kernel of the natural surjection from a graph product to the corresponding direct product. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of this yields conditions for a graph over which the graph product of arbitrary nontrivial groups (or some cyclic groups, or some finite groups) contains a hyperbolic surface group. In particular, the graph product of arbitrary nontrivial groups over a cycle of length at least five, or over its opposite graph, contains a hyperbolic surface group. For the case when the defining graphs have at most seven vertices, we completely characterize right-angled Coxeter groups with hyperbolic surface subgroups.