Boundaries and JSJ Decompositions of CAT(0)-Groups

Papasoglu, Panos; Swenson, Eric

doi:10.1007/s00039-009-0012-8

Cited by 48 publications

(70 citation statements)

References 29 publications

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“…See Bowditch [1], Bumagin, Kharlampovich and Miasnikov [2], Paulin [24] Scott and Swarup [26] and Papasoglu and Swenson [23] for other constructions of canonical JSJ splittings. As evidenced by the example on Figure 2, Theorems 4 and 5 do not hold for nonrelative JSJ splittings.…”

Section: Theoremmentioning

confidence: 99%

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: 99%

Trees of cylinders and canonical splittings

2011

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“…Here we can find some recent research on CAT(0) groups and their boundaries in [11], [13], [22], [27], [35], [36], [39], [41], [43], [44], [48] and [51]. Details of Coxeter groups and Coxeter systems are found in [6], [9] and [37], and details of Davis complexes which are CAT(0) spaces defined by Coxeter systems and their boundaries are found in [15], [16] and [47].…”

Section: Remarks and Questionsmentioning

confidence: 99%

On equivariant homeomorphisms of boundaries of CAT(0) groups and Coxeter groups

Hosaka

2015

Differential Geometry and its Applications

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Abstract. In this paper, we investigate an equivariant homeomorphism of the boundaries ∂X and ∂Y of two proper CAT(0) spaces X and Y on which a CAT(0) group G acts geometrically. We provide a sufficient condition and an equivalent condition to obtain a G-equivariant homeomorphism of the boundaries ∂X and ∂Y as a continuous extension of the quasi-isometry φ : Gx0 → Gy0 defined by φ(gx0) = gy0, where x0 ∈ X and y0 ∈ Y . In this paper, we say that a CAT(0) group G is equivariant (boundary) rigid, if G determines its ideal boundary by the equivariant homeomorphisms as above. As an application, we introduce some examples of (non-)equivariant rigid CAT(0) groups and we show that if Coxeter groups W1 and W2 are equivariant rigid as reflection groups, then so is W1 * W2. We also provide a conjecture on non-rigidity of boundaries of some CAT(0) groups.

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“…Several authors have expanded on these ideas, including [5,13,17,35,37], and, in a slightly different manner, [41]. In the group theoretic setting, JSJ decompositions encode a maximal amount of the information related to two-ended splittings (or, in the case of [38], Z-splittings) as a graph of groups.…”

Section: Jsj Decompositionsmentioning

confidence: 99%

“…We pause to note that even in the context of hyperbolic groups this strategy cannot be extended to produce different splittings from larger separating sets such as Cantor sets, [11]. An application for this construction to CAT(0) groups appears in [37]. Here, the group action on the CAT(0) boundary is analyzed via a variant of the convergence property to determine the isomorphism types of the vertex groups.…”

Section: Introductionmentioning

confidence: 99%