Groups and polytopes

TL;DR

This paper surveys the polytope invariant associated with certain groups, explaining its definition, properties, and open problems, with applications to 3-manifold groups, 2-generator 1-relator groups, and free-by-cyclic groups.

Contribution

It provides a comprehensive overview of the polytope invariant al(Gamma), including its definition, properties, and open questions in the context of group theory and topology.

Findings

The invariant al(Gamma) can be viewed as a polytope in many cases.

It applies to a wide class of groups including 3-manifold groups and free-by-cyclic groups.

The paper lists several open problems related to the invariant.

Abstract

In a series of papers the authors associated to an $L^2$-acyclic group $\Gamma$ an invariant $\mathcal{P}(\Gamma)$ that is a formal difference of polytopes in the vector space $H_1(\Gamma;\Bbb{R})$. This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic groups. In most of the above cases the invariant can be viewed as an actual polytope. In this survey paper we will recall the definition of the polytope invariant $\mathcal{P}(\Gamma)$ and we state some of the main properties. We conclude with a list of open problems.