A General Construction of JSJ Decompositions

Guirardel, Vincent; Levitt, Gilbert

doi:10.1007/978-3-7643-8412-8_6

Cited by 44 publications

(160 citation statements)

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“…We show in [18] that under the hypotheses of Theorem 6 the tree T c is maximal (for domination) among trees which are compatible with every other tree. In other words, T c belongs to the same deformation space as the JSJ compatibility tree defined in [18].…”

Section: Theoremmentioning

confidence: 94%

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Trees of cylinders and canonical splittings

2011

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Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depends on the deformation space of T; in particular, it is invariant under automorphisms of G if T is a JSJ splitting. We thus obtain Out(G)-invariant cyclic or abelian JSJ splittings. Furthermore, T_c has very strong compatibility properties (two trees are compatible if they have a common refinement).Comment: 38 pages, 2 figures. Reference updat

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Section: Theoremmentioning

confidence: 94%

“…This is used in [18] to construct (under suitable hypotheses) JSJ splittings of finitely generated groups, using acylindrical accessibility.…”

Section: Theoremmentioning

confidence: 99%

Trees of cylinders and canonical splittings

2011

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“…When all edge stabilizers are non-trivial, T may be viewed as the smallest common refinement (called lcm in [GL10]) of T and its tree of cylinders (see Subsection 2.2). Here is the construction ofT .…”

Section: Changing Tmentioning

confidence: 99%

McCool groups of toral relatively hyperbolic groups

Guirardel

Levitt

2015

Algebr. Geom. Topol.

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The outer automorphism group Out(G) of a group G acts on the set of conjugacy classes of elements of G. McCool proved that the stabilizer Mc(C) of a finite set of conjugacy classes is finitely presented when G is free. More generally, we consider the group Mc(H) of outer automorphisms Φ of G acting trivially on a family of subgroups H i , in the sense that Φ has representatives α i with α i equal to the identity on H i .When G is a toral relatively hyperbolic group, we show that these two definitions lead to the same subgroups of Out(G), which we call "McCool groups" of G. We prove that such McCool groups are of type VF (some finite index subgroup has a finite classifying space). Being of type VF also holds for the group of automorphisms of G preserving a splitting of G over abelian groups.We show that McCool groups satisfy a uniform chain condition: there is a bound, depending only on G, for the length of a strictly decreasing sequence of McCool groups of G. Similarly, fixed subgroups of automorphisms of G satisfy a uniform chain condition.

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“…There is an action of R + on X by scaling, the quotient is called a deformation space and denoted D. We [4] and independently Guirardel and Levitt [12], [13] have shown that for a finitely generated group, if the actions in D are irreducible and there is a reduced G-tree with finitely generated vertex stabilizers, then D is contractible. The topology for the preceding statement is the axes topology induced from the embedding D → RP C where C is the set of all conjugacy classes of elements in G, or equivalently the Gromov-Hausdorff topology.…”

Section: Deformation Spaces Of G-treesmentioning

confidence: 99%