#### Volume 8, issue 3 (2004)

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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 ${ℝ}^{n}$–trees. We first prove that Sela’s limit groups do have a free action on an ${ℝ}^{n}$–tree. We then prove that a finitely generated group having a free action on an ${ℝ}^{n}$–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
Primary: 20E08
Secondary: 20E26
##### Publication
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