Let G be a finitely generated group. Two simplicial G-trees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include Culler–Vogtmann's outer space and spaces of JSJ decompositions. We discuss what features are common to trees in a given deformation space, how to pass from one tree to all other trees in its deformation space, and the topology of deformation spaces. In particular, we prove that all deformation spaces are contractible complexes.
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Vincent Guirardel, Gilbert Levitt, Deformation spaces of trees. Groups Geom. Dyn. 1 (2007), no. 2, pp. 135–181DOI 10.4171/GGD/8