Finiteness properties of cubulated groups

Hruska, G. Christopher; Wise, Daniel T.

doi:10.1112/s0010437x13007112

Cited by 64 publications

(92 citation statements)

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“…The theory of hyperbolic groups acting properly and cocompactly on CAT(0) cube complexes is by now well developed (see [23,13,17,3,11], etc.). In the relatively hyperbolic situation, two generalizations have been previously studied: proper and cocompact actions (as in [22,23]) and proper and 'cosparse' actions (see [16,22]).…”

Section: Relative Cubulationsmentioning

confidence: 99%

Relative cubulations and groups with a 2-sphere boundary

Einstein¹,

Groves²

2020

Compositio Math.

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Section: Relative Cubulationsmentioning

confidence: 99%

Relative cubulations and groups with a 2-sphere boundary

Einstein¹,

Groves²

2020

Compositio Math.

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“…In this section, we explain how Corollary 1.3 follows from the preceding theorems and [9,30]. We begin by briefly reviewing the terminology associated to cube complexes and the groups that act on them.…”

Section: Cubulating the Fundamental Groupmentioning

confidence: 99%

“…Finally, in Section 8, we explain how to combine Theorems 1.1 and 1.2 with results of Bergeron-Wise [9] and Hruska-Wise [30] to show that M is homotopy equivalent to a compact cube complex. 1.…”

Section: Introductionmentioning

confidence: 99%

Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3–manifolds

Cooper

Futer

2019

Geom. Topol.

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This paper proves that every finite volume hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, non-asymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary of this result and a companion statement about surfaces with cusps, we recover Wise's theorem that the fundamental group of M acts freely and cocompactly on a CAT(0) cube complex.Hatcher showed that a compact manifold bounded by a torus has only finitely many embedded boundary slopes [27]. Hass, Rubinstein, and Wang [26] (refined by Zhang [53]) showed that there are only finitely many immersed boundary slopes whose surfaces have bounded Euler characteristic.Baker [4] gave the first example of a hyperbolic manifold with infinitely many immersed boundary slopes, while Baker and Cooper [5] showed that all slopes of even numerator in the figure-eight knot complement are virtual boundary slopes. Oertel [40] found a manifold with one cusp so that all slopes are immersed boundary slopes. Maher [36] gave many families, including all 2-bridge knots, for which every slope is an immersed boundary slope. Subsequently, Baker and Cooper [6, Theorem 9.4] showed that all slopes of one-cusped manifolds are immersed boundary slopes. Przytycki and Wise [42, Proposition 4.6] proved the same result for all slopes of multi-cusped hyperbolic manifolds.The surfaces constructed in those papers are not necessarily quasi-Fuchsian, because they may contain annuli parallel to the boundary. By contrast, Theorem 1.2 produces a ubiquitous collection of QF surfaces realizing every slope as an immersed boundary slope.1.1. Applications to cubulation and virtual problems. Theorems 1.1 and 1.2 have an application to the study of 3-manifold groups acting on CAT(0) cube complexes. We refer the reader to Section 8 for the relevant definitions.Theorems 1.1 and 1.2, combined with results of Bergeron and Wise [9] and Hruska and Wise [30], have the following immediate consequence.Corollary 1.3. Let M be a cusped hyperbolic 3-manifold. Choose a pair of distinct slopes α(V ), β(V ) for every cusp V ⊂ M . Then Γ = π 1 (M ) acts freely and cocompactly on a CAT(0) cube complex X dual to finitely many immersed quasi-Fuchsian surfaces S 1 , . . . , S k . Every surface S i is either closed or has immersed slope α(V ) or β(V ) for one cusp V ⊂ M .Thus M is homotopy equivalent to a compact non-positively curved cube complex X = X/Γ, whose immersed hyperplanes correspond to immersed quasi-Fuchsian surfaces S 1 , . . . , S k . Corollary 1.3 is not new. The statement that π 1 M acts freely and cocompactly on a cube complex X is an important theorem due to Wise [52]. In Wise's work, this result is obtained in the final step of his inductive construction of a virtual quasiconvex hierarchy for π 1 M . The purpose of this inductive construction is to establish ...

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“…Suppose that the group G acts by isometries on M , and W is G-invariant in the sense that gW ∈ W for each geometric wall W and each g ∈ G. Then G acts on the dual cube complex X. The collection W of walls satisfies the linear separation property if there exist constants K 1 , K 2 such that, for all p, q ∈ M , d(p, q) K 1 #(p, q) + K 2 , and it is shown in [18] that if G acts metrically properly on M and W satisfies the linear separation property, then G acts properly on X. In our situation, X is always locally finite, so that G acts metrically properly on X if and only if the stabilizer of each cube is finite.…”

Section: The Cube Complex Dual To a Wallspacementioning

confidence: 99%

Cocompactly cubulated crystallographic groups

Hagen

2014

Journal of the London Mathematical Society

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We prove that the simplicial boundary of a CAT(0) cube complex admitting a proper, cocompact action by a virtually Z n group is isomorphic to the hyperoctahedral triangulation of S n−1 , providing a class of groups G for which the simplicial boundary of a G-cocompact cube complex depends only on G. We also use this result to show that the cocompactly cubulated crystallographic groups in dimension n are precisely those that are hyperoctahedral. We apply this result to answer a question of Wise on cocompactly cubulating virtually free abelian groups.The main thrust of this example is that a cocompact cubulation of Z m × W would yield, via Theorem B, an action of W on a 3-cube with c acting as a 6-fold rotation; this is impossible. We also answer Wise's question negatively for the Dunbar example discussed in [29, Example 16.11]. On the other hand, we obtain the following as a consequence of Corollary 5.3, thus answering a weaker version of Wise's question affirmatively.Corollary C. Let G be an n-dimensional crystallographic group. Then there exists N n such that Z m G is a cocompactly cubulated crystallographic group for all m N − n and a suitably defined action of G on Z m .

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Finiteness properties of cubulated groups

Cited by 64 publications

References 50 publications

Relative cubulations and groups with a 2-sphere boundary

Relative cubulations and groups with a 2-sphere boundary

Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3–manifolds

Cocompactly cubulated crystallographic groups

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