CAT(0) spaces with polynomial divergence of geodesics

Macura, Nataša

doi:10.1007/s10711-012-9754-9

Cited by 20 publications

(34 citation statements)

References 9 publications

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“…In this appendix we discuss the relationship between our constructions of CAT(0) groups with divergence polynomial of any degree, and those of Macura [19].…”

Section: Appendix a Relationship With Examples Of Macuramentioning

confidence: 99%

“…For d ≥ 2, we denote by G d the group constructed in [19] with presentation We would like to use covering theory to investigate common finite index subgroups of G d and W d . Any finite index subgroup of G d is the fundamental group of a finite square complex Q such that there is a combinatorial covering map Ψ : Q → X d .…”

Section: Appendix a Relationship With Examples Of Macuramentioning

confidence: 99%

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Divergence in right-angled Coxeter groups

Dani

Thomas

2014

Trans. Amer. Math. Soc.

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Abstract. Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex. IntroductionThe divergence of a pair of geodesics is a classical notion related to curvature. Roughly speaking, given a pair of geodesic rays emanating from a basepoint, their divergence measures, as a function of r, the length of a shortest "avoidant" path connecting their time-r points. A path is avoidant if it stays at least distance r away from the basepoint. In [15], Gersten used this idea to define a quasi-isometry invariant of spaces, also called divergence. We recall the definitions of both notions of divergence in Section 2.The divergence of every pair of geodesics in Euclidean space is a linear function, and it follows from Gersten's definition that any group quasi-isometric to Euclidean space has linear divergence. In a δ-hyperbolic space, any pair of non-asymptotic rays diverges exponentially; thus the divergence of any hyperbolic group is exponential. In symmetric spaces of non-compact type, the divergence is either linear or exponential, and Gromov suggested in [16] the same should be true in CAT(0) spaces.Divergence has been investigated for many important groups and spaces, and contrary to Gromov's expectation, quadratic divergence is common. Gersten first exhibited quadratic divergence for certain CAT(0) spaces in [15]. He then proved in [14] that the divergence of the fundamental group of a closed geometric 3-manifold is either linear, quadratic or exponential, and characterised the (geometric) ones with quadratic divergence as the fundamental groups of graph manifolds. showed that all graph manifold groups have quadratic divergence. More recently, Duchin-Rafi [13] established that the divergence of Teichmüller space and the mapping class group is quadratic (for mapping class groups this was also obtained by Behrstock in [5]). Druţu-Mozes-Sapir [12] have conjectured that the divergence of lattices in higher rank semisimple Lie groups is always linear, and proved this conjecture in some cases. Abrams et al [1] and independently Behrstock-Charney [2] have shown that if A Γ is the right-angled Artin group associated to a graph Γ, the group A Γ has either linear or quadratic divergence, and its divergence is linear if and only if Γ is (the 1-skeleton of) a join.In this work we study the divergence of 2-dimensional right-angled Coxeter groups. Our first main result is Theorem 1.1 below, which characterises such groups with linear and quadratic divergence in terms of their defining graphs. This result can be seen as a step in the quasi-isometry classification of (right-angled) Coxeter groups, about which very little is known.We note that by [10], every right-angled Artin group is a finite index subgroup of, and therefore quasi-isometric to, a right-angled Coxeter group. However, in contrast to the setting of right-angled Artin gr...

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“…In this appendix we discuss the relationship between our constructions of CAT(0) groups with divergence polynomial of any degree, and those of Macura [19].…”

Section: Appendix a Relationship With Examples Of Macuramentioning

confidence: 99%

Section: Appendix a Relationship With Examples Of Macuramentioning

confidence: 99%

Divergence in right-angled Coxeter groups

Dani

Thomas

2014

Trans. Amer. Math. Soc.

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“…More recently, there have been some examples of CAT(0) groups exhibiting polynomial divergence of any positive integer degree [BD14] [BH16] [DT15] [Mac13]. Additionally, there are even constructions of more exotic infinitely presented groups (not necessarily CAT(0)) with divergence function not a polynomial [GS] [OOS09].…”

Section: Introductionmentioning

confidence: 99%

Divergence of CAT(0) cube complexes and Coxeter groups

Levcovitz

2018

Algebr. Geom. Topol.

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We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. As another application, we provide an inductive graph theoretic criterion on a right-angled Coxeter group's defining graph which allows us to recognize arbitrary integer degree polynomial divergence for many infinite classes of right-angled Coxeter groups. We also provide similar divergence results for some classes of Coxeter groups which are not right-angled.

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“…Upper divergence is more diverse since the upper divergence of a finitely generated group can be any polynomial or exponential function (see Macura [Mac13] and Sisto [Sis]). Upper divergence has been studied by Macura [Mac13], Behrstock-Charney [BC12], Duchin-Rafi [DR09], Druţu-Mozes-Sapir [DMS10], Sisto [Sis] and others. Moreover, upper divergence is a quasi-isometry invariant, and it is therefore a useful tool to classify finitely generated groups up to quasi-isometry.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, we only need to choose K to be a one-ended hyperbolic group to have the upper relative divergence Div(G, H) as the exponential function. Similarly, we can choose a one-ended group K such that Div(K, e) is equivalent to a desired polynomial (for example, see [Mac13]) and Div(G, H) is also equivalent to this desired polynomial.…”

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confidence: 99%

Relative divergence of finitely generated groups

Tran

2015

Algebr. Geom. Topol.

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Abstract. We generalize the concept of divergence of finitely generated groups by introducing the upper and lower relative divergence of a finitely generated group with respect to a subgroup. Upper relative divergence generalizes Gersten's notion of divergence, and lower relative divergence generalizes a definition of Cooper-Mihalik. While the lower divergence of Cooper-Mihalik can only be linear or exponential, relative lower divergence can be any polynomial or exponential function. In this paper, we examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of CAT(0) groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups.

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CAT(0) spaces with polynomial divergence of geodesics

Abstract: We construct a family of finite 2-complexes whose universal covers are CAT(0) and have polynomial divergence of desired degree. This answers a question of Gersten, namely whether such CAT(0) complexes exist

Cited by 20 publications

References 9 publications

Divergence in right-angled Coxeter groups

Divergence in right-angled Coxeter groups

Divergence of CAT(0) cube complexes and Coxeter groups

Relative divergence of finitely generated groups

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