---
abstract: 'We give a simple proof of the finite presentation of Sela’s limit groups by using free actions on ${{\mathbb{R}}}^n$–trees. We first prove that Sela’s limit groups do have a free action on an ${{\mathbb{R}}}^n$–tree. We then prove that a finitely generated group having a free action on an ${{\mathbb{R}}}^n$–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.'
address: |
    Laboratoire E. Picard, UMR 5580, Bât 1R2\
    Université Paul Sabatier, 118 rte de Narbonne\
    31062 Toulouse cedex 4, France
author:
- Vincent Guirardel
title: 'Limit groups and groups acting freely on ${{\mathbb{R}}}^n$–trees'
---

Introduction
============

Limit groups have been introduced by Z Sela in the first paper of his solution of Tarski’s problem [@Sela_diophantine1]. These groups appeared to coincide with the long-studied class of finitely generated fully residually free groups (see [@Baumslag_residually], [@Baumslag_generalised], [@KhMy_irreducible1; @KhMy_irreducible2], [@Chi_introduction] and references).

A limit group is a limit of free groups in the space of marked groups. 