Volume 19, issue 2 (2015)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Limit groups over partially commutative groups and group actions on real cubings

Montserrat Casals-Ruiz and Ilya Kazachkov

Geometry & Topology 19 (2015) 725–852

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  U of finitely generated fully residually partially commutative groups.

Our first main goal is to give an algebraic characterisation of the class  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 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 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.

equations in groups, partially commutative group, right-angled Artin group, cube complexes and generalisations, group actions
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
Secondary: 20F70, 20E08
Received: 2 May 2012
Revised: 5 April 2014
Accepted: 7 June 2014
Published: 10 April 2015
Proposed: Walter Neumann
Seconded: Mikhail Gromov, Yasha Eliashberg
Montserrat Casals-Ruiz
Mathematical Institute
University of Oxford
Radcliffe Observatory Quarter
Woodstock Road
Oxford OX2 6GG
Ilya Kazachkov
Department of Mathematics
University of the Basque Country UPV/EHU
Barrio Sarriena
48940 Leioa, Vizcaya
Basque Foundation for Science