Volume 8, issue 3 (2004)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Limit groups and groups acting freely on $\mathbb{R}^n$–trees

Vincent Guirardel

Geometry & Topology 8 (2004) 1427–1470

arXiv: math.GR/0306306

Abstract

We give a simple proof of the finite presentation of Sela’s limit groups by using free actions on n–trees. We first prove that Sela’s limit groups do have a free action on an n–tree. We then prove that a finitely generated group having a free action on an n–tree can be obtained from free abelian groups and surface groups by a finite sequence of free products and amalgamations over cyclic groups. As a corollary, such a group is finitely presented, has a finite classifying space, its abelian subgroups are finitely generated and contains only finitely many conjugacy classes of non-cyclic maximal abelian subgroups.

Keywords
$\mathbb{R}^n$–tree, limit group, finite presentation
Mathematical Subject Classification 2000
Primary: 20E08
Secondary: 20E26
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Publication
Received: 14 October 2003
Revised: 26 November 2004
Accepted: 29 September 2004
Published: 27 November 2004
Proposed: Martin Bridson
Seconded: Benson Farb, Walter Neumann
Authors
Vincent Guirardel
Laboratoire E. Picard
UMR 5580
Bât 1R2
Université Paul Sabatier
118 rte de Narbonne
31062 Toulouse cedex 4
France