Tits alternatives for graph products

TL;DR

This paper explores the Tits Alternative in graph products of groups, establishing conditions under which subgroups satisfy certain properties, and introduces a theory of parabolic subgroups to analyze subgroup structures.

Contribution

It proves that the Tits Alternative for graph products depends on vertex groups and develops a new framework of parabolic subgroups for subgroup analysis.

Findings

Finitely generated subgroups of virtually solvable graph products are either virtually solvable or large.

Non-abelian subgroups of right angled Artin groups map onto free groups of rank 2.

Group properties are stable under graph product constructions.

Abstract

We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the theory of parabolic subgroups, which allows to describe the structure of subgroups of graph products that contain no non-abelian free subgroups. We also obtain a number of results regarding the stability of some group properties under taking graph products.