Geometry & Topology

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

Vincent Guirardel

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We give a simple proof of the finite presentation of Sela’s limit groups by using free actions on n–trees. We first prove that Sela’s limit groups do have a free action on an n–tree. We then prove that a finitely generated group having a free action on an 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.

Article information

Geom. Topol., Volume 8, Number 3 (2004), 1427-1470.

Received: 14 October 2003
Revised: 26 November 2004
Accepted: 29 September 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 20E26: Residual properties and generalizations; residually finite groups

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


Guirardel, Vincent. Limit groups and groups acting freely on $\mathbb{R}^n$–trees. Geom. Topol. 8 (2004), no. 3, 1427--1470. doi:10.2140/gt.2004.8.1427.

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