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

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.