The minimum $b_2$ problem for right-angled Artin groups

Abstract

This paper focuses on tools for constructing 4-manifolds that have fundamental group $G$ isomorphic to a right-angled Artin group and that are also minimal, in the sense that they minimize $b_2(M)$, the dimension of $H_2(M;\mathbb{Q})$. For a finitely presented group $G$, define $h(G)=\min \{b_2(M)|M\in \mathcal{M}(G)\}$. In this paper, we explore the ways in which we can bound $h(G)$ from below using group cohomology and the tools necessary to build 4-manifolds that realize these lower bounds. We give solutions for right-angled Artin groups, or RAAGs, when the graph associated to $G$ has no 4-cliques, and further we reduce this problem to the case when the graph is connected and contains only 4-cliques. We then give solutions for many infinite families of RAAGs and provide a conjecture to the solution for all RAAGs.

40 pages, 46 figures, appendix containing Sage code

References in corpus (1)

• Minimizing Euler characteristics of symplectic four-manifolds

Cited by in corpus (1)

• Topological 4-manifolds with right-angled Artin fundamental groups