Limit groups and groups acting freely on ℝn–trees

Guirardel, Vincent¹

¹ Laboratoire E. Picard, UMR 5580, Bât 1R2, Université Paul Sabatier, 118 rte de Narbonne, 31062 Toulouse cedex 4, France

Geometry & topology, Tome 8 (2004) no. 3, pp. 1427-1470.

Résumé

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.

Détail

DOI : 10.2140/gt.2004.8.1427

Keywords: $\mathbb{R}^n$–tree, limit group, finite presentation

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