Local connectivity of right-angled Coxeter group boundaries

Ruane, Kim; Tschantz, Steve

doi:10.1515/jgt.2007.042

Cited by 16 publications

(15 citation statements)

References 7 publications

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“…Intuitively, the next result says that when a Coxeter group acts geometrically on a CAT(0) space, CAT(0) geodesics are tracked by Cayley graph geodesics. This result generalizes the right-angled version of the same result in [5]. Corollary 6.3.…”

Section: Consequences Of the Main Theoremsupporting

confidence: 80%

“…Proof. We use an elementary form of a construction of a 'filter' in [5] (where a partial classification of right-angled Coxeter groups with locally connected boundaries is produced). Suppose that W acts geometrically on the CAT(0) space X, with base point x.…”

Section: Consequences Of the Main Theoremmentioning

confidence: 99%

“…Clearly, the algebraic properties of G are far more apparent in Cayley geodesics than in CAT(0) geodesics. This theme is highlighted in [5] where local connectivity of boundaries of right-angled Coxeter groups are analyzed.…”

Section: Introductionmentioning

confidence: 99%

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Geodesically tracking quasi-geodesic paths for Coxeter groups

Tschantz¹

2013

Bulletin of the London Mathematical Society

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If Λ is the Cayley graph of a Gromov hyperbolic group, then it is a fundamental fact that quasi‐geodesics in Λ are tracked by geodesics. Let (W, S) be a finitely generated Coxeter system and Λ be the Cayley graph of (W, S). For general Coxeter groups, not all quasi‐geodesic rays in Λ are tracked by geodesics. In this paper, we classify the Λ‐quasi‐geodesic rays that are tracked by geodesics. As corollaries we show that if W acts geometrically on a CAT(0) space X, then CAT(0) geodesics in X are tracked by Cayley graph geodesics (taking the Cayley graph as equivariantly placed in X) and for any A ⊂ S, the special subgroup 〈A〉 is quasi‐convex in X. We also show that if g is an element of infinite order for (W, S), then the subgroup 〈g〉 is tracked by a Cayley geodesic in Λ (in analogy with the corresponding result for word hyperbolic groups).

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Section: Consequences Of the Main Theoremsupporting

confidence: 80%

Section: Consequences Of the Main Theoremmentioning

confidence: 99%

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Geodesically tracking quasi-geodesic paths for Coxeter groups

Tschantz¹

2013

Bulletin of the London Mathematical Society

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“…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]. We can find some recent research on boundaries of Coxeter groups in [10], [17], [18], [19], [33], [40]. Here we note that every cocompact discrete reflection group of a geodesic space becomes a Coxeter group [32], and if a Coxeter group W is a cocompact discrete reflection group of a CAT(0) space X, then the CAT(0) space X has a structure similar to the Davis complex [34].…”

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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“…5 seems to give some promise for constructing a word hyperbolic group that is simply connected at infinity, but with non-simply connected boundary. See [12][13][14] for a discussion of Coxeter groups and general CAT(0) groups with non-locally connected boundary.…”

Section: Questionmentioning

confidence: 99%

Homotopy of ends and boundaries of CAT(0) groups

Conner

Tschantz

2006

Geom Dedicata

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For n ≥ 0, we exhibit CAT(0) groups that are n-connected at infinity, and have boundary which is (n − 1)-connected, but this boundary has non-trivial nth-homotopy group. In particular, we construct 1-ended CAT(0) groups that are simply connected at infinity, but have a boundary with non-trivial fundamental group.Our base examples are 1-ended CAT(0) groups that have non-path connected boundaries. In particular, we show all parabolic semidirect products of the free group of rank 2 and Z have a non-path connected boundary.

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Local connectivity of right-angled Coxeter group boundaries

Cited by 16 publications

References 7 publications

Geodesically tracking quasi-geodesic paths for Coxeter groups

Geodesically tracking quasi-geodesic paths for Coxeter groups

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

Homotopy of ends and boundaries of CAT(0) groups

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