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Limit groups and groups acting freely on $$\mathbb{R}^n$$-trees. (English) Zbl 1114.20013
A ‘limit group’ is a limit of free groups in the space of marked groups. A theorem due to Kharlampovich-Myasnikov, Pfander and Sela states that a limit group is inductively obtained from free Abelian groups and surface groups by taking free products and amalgamations over $$\mathbb{Z}$$. This implies that such a group is finitely presented, that it has a finite classifying space, that its Abelian subgroups are finitely generated and that it contains only finitely many conjugacy classes of non-cyclic maximal Abelian subgroups.
In the paper under review, the author gives another proof of the fact that a limit group is inductively obtained from free Abelian groups and surface groups by taking free products and amalgamations over $$\mathbb{Z}$$. He first proves that a limit group acts freely on an $$\mathbb{R}^n$$-tree, and he then proves that a finitely generated group acting freely on an $$\mathbb{R}^n$$-tree can be obtained from free Abelian groups and from surface groups by a finite sequence of free products and amalgamation over cyclic groups.

##### MSC:
 20E08 Groups acting on trees 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E26 Residual properties and generalizations; residually finite groups 20F05 Generators, relations, and presentations of groups 20E05 Free nonabelian groups 57M07 Topological methods in group theory
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