Minimal volume entropy of free-by-cyclic groups and 2-dimensional right-angled Artin groups

TL;DR

This paper characterizes when free-by-cyclic groups and 2-dimensional right-angled Artin groups have minimal volume entropy zero, providing algebraic and geometric criteria, and establishes positive lower bounds in the nonvanishing case.

Contribution

It offers new algebraic and geometric criteria for minimal volume entropy vanishing and bounds for these specific classes of groups, extending understanding of their geometric properties.

Findings

Characterization of zero minimal volume entropy for the groups

Positive lower bounds for nonvanishing minimal volume entropy

A criterion based on fiber π₁-growth collapse and non-collapsing assumptions

Abstract

Let $G$ be a free-by-cyclic group or a 2-dimensional right-angled Artin group. We provide an algebraic and a geometric characterization for when each aspherical simplicial complex with fundamental group isomorphic to $G$ has minimal volume entropy equal to 0. In the nonvanishing case, we provide a positive lower bound to the minimal volume entropy of an aspherical simplicial complex of minimal dimension for these two classes of groups. Our results rely upon a criterion for the vanishing of the minimal volume entropy for 2-dimensional groups with uniform uniform exponential growth. This criterion is shown by analyzing the fiber $\pi_1$-growth collapse and non-collapsing assumptions of Babenko-Sabourau.