Graph braid groups and right-angled Artin groups

Abstract

We give a sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\geq5$. And prove that if a graph that is not $\It{T_3}$, contains $\It{T_0}$ but does not contain $\It{S_0}$ as topological subgraph, then graph braid group is not a right-angled Artin group for braid index $\geq 4$. In chapter 1, we introduce the basic notions and well-known facts about the relationship between graph braid groups and right-angled Artin groups. In chapter 2, we explain how to simplify configuration spaces using the discrete Morse theory. As applications, we obtain a presentation of the graph braid group for any graph containing neither $\It{T_0}$ nor $\It{S_0}$ as topological subgraph and prove that the graph braid group is a right-angled Artin group. In chapter 3, we first prove that for a graph not containing $\It{S_0}$ as topological subgraph, the homology groups of graph braid groups are free abelian groups. And for low braid indices, we compute cohomology algebras of graph braid groups of graphs that are simple enough and prove that graph braid groups are not right-angled Artin groups. Thus, by using that graphs and a homomorphism of cohomology algebras of graph braid groups that is induced by the embedding of graphs, we prove that for a graph that is not $\It{T_3}$ containing $\It{T_0}$but not containing $\It{S_0}$ as topological subgraph, the graph braid group is not a right-angled Artin group for braid index $\geq 4$.