Group Actions on Real Cubings and Limit Groups over Partially Commutative Groups

TL;DR

This paper introduces real cubings, generalizes the concept of real trees, and characterizes groups acting on them as subgroups of graph towers, extending classical results on free and surface groups to partially commutative groups.

Contribution

It defines real cubings and characterizes groups acting on them as subgroups of graph towers, extending Rips' theorem and limit group classifications.

Findings

Groups acting on real cubings are subgroups of graph towers.

Characterization of limit groups over partially commutative groups.

Generalization of classical theorems on free and surface groups.

Abstract

We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation of real trees. Our main result states that a finitely generated group $G$ acts nicely (essentially freely and co-specially) on a real cubing if and only if it is a subgroup of a graph tower (a higher dimensional generalisation of $\omega$-residually free towers and NTQ-groups). It follows that $G$ acts freely, essentially freely and co-specially on a real cubing if and only if $G$ is a subgroup of the graph product of cyclic and (non-exceptional) surface groups. In the particular case when the real cubing is a tree, it follows that $G$ acts freely, essentially freely and co-specially on the real cubing if and only if it is a subgroup of the free product of abelian and surface groups. Hence, our main result can be regarded as a generalisation of the Rips' theorem on free actions on real trees. We apply our results to obtain a characterisation of limit groups over partially commutative groups as subgroups of graph towers. This result generalises the work of Kharlampovich-Miasnikov, \cite{KhMNull}, Sela, \cite{Sela1} and Champetier-Guirardel, \cite{CG} on limit groups over free groups.