A cut point theorem for $\rm{CAT}(0)$ groups

Swenson, Eric

doi:10.4310/jdg/1214425538

Cited by 58 publications

(93 citation statements)

References 19 publications

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“…By [28] A ⊂ ΛMin(h i ) = ΛZ h i for all i. Since Z h i is convex, we have by [28] that the centralizer of H, Z H = ∩Z h i is convex and …”

Section: Proofmentioning

confidence: 99%

“…If h ∈ G is hyperbolic and the centralizer Z h n is virtually cyclic for all n, then for some n, stab(h ± ) = Z h n Proof. By [28,Zipper Lemma] hyperbolic elements with the same endpoints have a common power. Thus we may assume that h has the minimal translation length of any hyperbolic element with endpoints h ± .…”

Section: Cat(0) Groups and Boundariesmentioning

confidence: 99%

“…Suppose not, then from [28,Theorem 17] there is a convex set Y ⊂ U with dim ∂Y < dim ∂U such that Y is invariant under the action of H and the action of H on Y can be extended to a geometric action on Y . Since dim ∂U < ∞ ( [28]), we get a contradiction by induction on boundary dimension (In the case where the boundary is zero dimensional, any group acting geometrically is virtually free.) Thus there exists h ∈ H hyperbolic.…”

Section: Cat(0) Groups and Boundariesmentioning

confidence: 99%

“…We extend the interval relation of R to T in the obvious way (as in [28]), so that in T , (x, y) = I x,y . It is clear that T is a complete pretree with no adjacent elements.…”

Section: Trees and Continuamentioning

confidence: 99%

“…However the question whether the boundaries have cut points also makes sense in this case. Indeed the second author showed that boundaries have no cut points under the assumption that the group does not contain an infinite torsion subgroup ( [28]). …”

Section: Introductionmentioning

confidence: 99%

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Boundaries and JSJ Decompositions of CAT(0)-Groups

Papasoglu¹,

Swenson

2009

Geom. Funct. Anal.

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Abstract. Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that ∂X has no cut points and that one can detect splittings of G over two-ended groups and recover its JSJ decomposition from ∂X.We show that any discrete action of a group G on a CAT(0) space X satisfies a convergence type property. This is used in the proof of the results above but it is also of independent interest. In particular, if G acts co-compactly on X, then one obtains as a Corollary that if the Tits diameter of ∂X is bigger than 3π 2 then it is infinite and G contains a free subgroup of rank 2.

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“…By [28] A ⊂ ΛMin(h i ) = ΛZ h i for all i. Since Z h i is convex, we have by [28] that the centralizer of H, Z H = ∩Z h i is convex and …”

Section: Proofmentioning

confidence: 99%

Section: Cat(0) Groups and Boundariesmentioning

confidence: 99%

Section: Cat(0) Groups and Boundariesmentioning

confidence: 99%

“…We extend the interval relation of R to T in the obvious way (as in [28]), so that in T , (x, y) = I x,y . It is clear that T is a complete pretree with no adjacent elements.…”

Section: Trees and Continuamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Boundaries and JSJ Decompositions of CAT(0)-Groups

Papasoglu¹,

Swenson

2009

Geom. Funct. Anal.

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Ends, Shapes, and Boundaries in Manifold Topology and Geometric Group Theory

Guilbault

2016

Springer Proceedings in Mathematics &Amp; Statistics

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This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often contractible, so distinguishing one from another requires techniques beyond the standard tools of algebraic topology. One approach uses end invariants, such as the number of ends or the fundamental group at infinity. Another approach seeks nice compactifications, then analyzes the boundaries. A thread connecting the two approaches is shape theory.In these notes we provide a careful development of several topics: homotopy and homology properties and invariants for ends of spaces, proper maps and homotopy equivalences, tameness conditions, shapes of ends, and various types of Z-compactifications and Z-boundaries. Classical and current research from both manifold topology and geometric group theory provide the context. Along the way, several open problems are encountered. Our

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Commensurators of abelian subgroups in CAT(0) groups

Huang

Prytuła

2019

Math. Z.

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We study the structure of the commensurator of a virtually abelian subgroup H in G, where G acts properly on a CAT(0) space X. When X is a Hadamard manifold and H is semisimple, we show that the commensurator of H coincides with the normalizer of a finite index subgroup of H. When X is a CAT(0) cube complex or a thick Euclidean building and the action of G is cellular, we show that the commensurator of H is an ascending union of normalizers of finite index subgroups of H. We explore several special cases where the results can be strengthened and we discuss a few examples showing the necessity of various assumptions. Finally, we present some applications to the constructions of classifying spaces with virtually abelian stabilizers.2000 Mathematics Subject Classification. Primary 20F65; Secondary 20F67.

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A cut point theorem for $\rm{CAT}(0)$ groups

Cited by 58 publications

References 19 publications

Boundaries and JSJ Decompositions of CAT(0)-Groups

Boundaries and JSJ Decompositions of CAT(0)-Groups

Ends, Shapes, and Boundaries in Manifold Topology and Geometric Group Theory

Commensurators of abelian subgroups in CAT(0) groups

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