Deformation spaces of trees

By Vincent Guirardel, Gilbert Levitt. 2006

85 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8fbf0f16a58813a6f050195e645d8289167dce99

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Citations in the corpus

The outer space of a free product (2005, methodology): “n, because selecting the unique point of T ﬁxed by G1 provides a continuous choice of a basepoint in Tas Tvaries in O. Because of this, contractibility of Culler-Vogtmann space is not proved here. In [15], we prove contractibility of arbitrary deformation spaces, using a general basepoint argument (”
Homotopy type of the complex of free factors of a free group (2018, methodology): “n F, the poset of free splittings FS n(A) of F relative to Ais contractible. Proof. Up to projectivising, FS n(A) is a subset of D. Let ˆ: kFS n(A)k [0;1] !D be the restriction of the map dened in [GL07a]. We will show that the image of ˆ is contained in kFS n(A)kand that the map is continuous with ”
The Isomorphism Problem for All Hyperbolic Groups (2010, methodology): “bounded Fuchsian group without reflection, and since � Ti� is Z-universally elliptic, it is either finite or conjugate into a boundary subgroup of Gv ([ GuL3 ]).”
Automorphisms of 2–dimensional right-angled Artin groups (2006, background): “The proof follows ideas of Skora [21] and Guirardel and Levitt [16; 15] on “unfolding” trees.”
METRIC PROPERTIES OF OUTER SPACE (2008, background): “Proof: Note that the Gromov topology, the Axes topology and the Weak topology coincide for Culler-Vogtmann Space, and a fuller discussion may be found here, [10].”
Hyperbolicity of the complex of free factors (2011, background): “Skora’s preprint was never published; the interested reader may find the details in [8] and [12].”
Trees of cylinders and canonical splittings (2008, background): “Having the same elliptic subgroups is an equivalence relation on the set of trees, the equivalence classes are called deformation spaces [9; 14].”
On the automorphism group of generalized Baumslag–Solitar groups (2005, background): “Assuming it is, consider (as in [ 16 ]) the set of groups of the form Z.a2/\K , where K is an elliptic subgroup of T not contained in Z.a2/.”