Limit groups over partially commutative groups and group actions on real cubings

Abstract

The study of limit groups, that is, finitely generated fully residually free groups, was a key first step towards the understanding of the elementary theory of a free group. In this paper we conduct a systematic study of the class $\mathfrak{U}$ of finitely generated fully residually partially commutative groups.

Our first main goal is to give an algebraic characterisation of the class $\mathfrak{U}$: a finitely generated group $G$ is fully residually partially commutative if and only if it is a subgroup of a graph tower (a group built hierarchically using partially commutative groups and (nonexceptional) surfaces.) Furthermore, if the group $G$ is given by its finite radical presentation, then the graph tower and the embedding can be effectively constructed. This result generalises the work of Kharlampovich and Miasnikov on fully residually free groups.

Following Sela’s approach to limit groups, the second goal of the paper is to provide a dynamical characterisation of the class $\mathfrak{U}$. We introduce a class of spaces, called real cubings, as higher-dimensional generalisations of real trees and show that a specific type of action on these spaces characterises the class $\mathfrak{U}$: a finitely generated group acts freely cospecially on a real cubing if and only if it is fully residually partially commutative. As a corollary we get that (geometric) limit groups over partially commutative groups are fully residually partially commutative. This result generalises the work of Sela on limit groups over free groups.