Limit groups and groups acting freely on $\bbR^n$-trees

TL;DR

This paper provides a simplified proof that Sela's limit groups are finitely presented by demonstrating their free actions on bR^n-trees and characterizing groups with such actions.

Contribution

It introduces a straightforward proof of the finite presentation of Sela's limit groups using free actions on bR^n-trees and characterizes groups with these actions.

Findings

Sela's limit groups act freely on bR^n-trees.

Groups with free bR^n-tree actions are constructed from abelian and surface groups.

Such groups are finitely presented with finitely generated abelian subgroups.

Abstract

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