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.

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Vincent Guirardel
Laboratoire E. Picard, UMR 5580, Bat 1R2
Universite Paul Sabatier, 118 rte de Narbonne
31062 Toulouse cedex 4, France

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