#### Geometry & Topology, Vol. 8 (2004)
Paper no. 39, pages 1427--1470.

## Limit groups and groups acting freely on R^n-trees

### Vincent Guirardel

**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**.
R^n-tree, limit group, finite presentation

**AMS subject classification**.
Primary: 20E08.
Secondary: 20E26.

**DOI:** 10.2140/gt.2004.8.1427

**E-print:** `arXiv:math.GR/0306306`

Submitted to GT on 14 October 2003.
(Revised 26 November 2004.)
Paper accepted 29 September 2004.
Paper published 27 November 2004.

Notes on file formats
Vincent Guirardel

Laboratoire E. Picard, UMR 5580, Bat 1R2

Universite Paul Sabatier, 118 rte de Narbonne

31062 Toulouse cedex 4, France

Email: guirardel@picard.ups-tlse.fr

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