Limit groups and groups acting freely on ℝn–trees

Guirardel, Vincent

doi:10.2140/gt.2004.8.1427

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Limit groups and groups acting freely on $\mathbb{R}^n$–trees

Vincent Guirardel

Geometry & Topology 8 (2004) 1427–1470

DOI: 10.2140/gt.2004.8.1427

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

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

Volume 8, issue 3 (2004)