On the linearity of fundamental groups of rose and star graphs of groups

TL;DR

This paper investigates when the fundamental groups of certain graphs of groups, specifically rose and star graphs with various vertex groups, are linear over the integers, providing new conditions for their linearity.

Contribution

It establishes the linearity of these fundamental groups over or a broad class of graphs of groups with specific vertex and edge group conditions.

Findings

Proves linearity over or fundamental groups of rose and star graphs with free, free abelian, or RAAG vertex groups.

Identifies conditions on edge groups that ensure linearity.

Extends known results to new classes of graphs of groups.

Abstract

Let $G$ be a fundamental group of a graph of group where the graph is a rose or a star graph and the vertex groups are free groups, free abelian groups or right-angled Artin groups. We prove the linearity of $G$ over $\mathbb{Z}$ under various conditions on its edge groups.