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Deformation spaces of trees. (English) Zbl 1134.20026
Let \(G\) be a finitely generated group. A \(G\)-tree is a simplicial metric tree on which \(G\) acts by isometries. Two such actions are considered equivalent modulo equivariant isometry. A subgroup \(H\subset G\) is said to be elliptic if it fixes a point. The notion of a deformation space for such actions was introduced by M. Forester [in his paper in Geom. Topol. 6, 219-267 (2002; Zbl 1118.20028)]. By definition, two \(G\)-trees are in the same deformation space if they have the same elliptic subgroups. Forester described some moves (“elementary moves”) such that any two \(G\)-actions that are in the same deformation space can be joined by a finite sequence of such moves. Examples of deformation spaces include the Culler-Vogtmann outer space, spaces of JSJ decompositions and spaces constructed by McCullough-Miller to study automorphisms of free products.
In the paper under review, the authors discuss general properties of deformation spaces. They study two topologies on such a space, namely, the Hausdorff-Gromov topology and the weak topology associated to a natural cell-complex structure of the deformation space. Using ideas of R. Skora, they prove that all deformation spaces are contractible in the weak topology. The paper contains other results, in particular concerning the following natural questions: Given a tree, what are the moves that are needed in order to generate all trees in its deformation space? Are slide moves sufficient? What is common to trees in a given deformation space? In particular, to what extent do trees in the same deformation space have the same edge and vertex stabilizers? Is a deformation space finite-dimensional? Does it have a finite-dimensional spine?

20E08 Groups acting on trees
57M07 Topological methods in group theory
20F65 Geometric group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E36 Automorphisms of infinite groups
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