Cocompactly cubulated 2-dimensional Artin groups

Huang, Jingyin; Jankiewicz, Kasia; Przytycki, Piotr

doi:10.4171/cmh/394

Cited by 25 publications

(13 citation statements)

References 32 publications

(30 reference statements)

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“…Given a hyperbolic g ∈ Aut(X), we describe combinatorially a certain invariant subcomplex associated to g which consists of all the lines parallel to axes in G. (This subcomplex is discussed as well in [11] and is slightly different than the minimal set of G, as described in [3] or [10]. )…”

Section: The Parallel Subset Of An Elementmentioning

confidence: 99%

Uniform exponential growth for CAT(0) square complexes

Kar

Sageev

2019

Algebr. Geom. Topol.

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In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if F is a finite collection of hyperbolic automorphisms of a CAT(0) square complex X, then either there exists a pair of words of length at most 10 in F which freely generate a free semigroup, or all elements of F stabilize a flat (of dimension 1 or 2 in X). As a corollary, we obtain a lower bound for the growth constant, 10 √ 2, which is uniform not just for a given group acting freely on a given CAT(0) cube complex, but for all groups which are not virtually abelian and have a free action on a CAT(0) square complex.

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Section: The Parallel Subset Of An Elementmentioning

confidence: 99%

Uniform exponential growth for CAT(0) square complexes

Kar

Sageev

2019

Algebr. Geom. Topol.

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“…• Artin groups that can be cocompactly cubulated. An important class of such Artin groups is the class of right-angled Artin groups, namely Artin groups such that m st = 2 or ∞ for every s = t ∈ S. Beyond them, a few classes of Artin groups have been shown to be cocompactly cubulated [HJP16,Hae17], but the conjectural picture states that the class of cocompactly cubulated Artin groups is extremely constrained [Hae17]. • Artin groups of finite type (also known as spherical Artin groups), i.e.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Acylindrical actions for two-dimensional Artin groups of hyperbolic type

Martin¹,

Przytycki²

2019

Preprint

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For a two-dimensional Artin group A whose associated Coxeter group is hyperbolic, we prove that the action of A on the hyperbolic space obtained by coning off certain subcomplexes of its modified Deligne complex is acylindrical and universal. As a consequence, we obtain the Tits alternative for A, and we classify its subgroups that virtually split as a direct product. We also extend the classification of maximal virtually abelian subgroups to arbitrary two-dimensional Artin groups. A key ingredient in our approach is a simple criterion to show the acylindricity of an action on a two-dimensional CAT(−1) complex.

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“…The stabilisers of the vertices of Φ, appearing in (i), are cyclic or conjugates of the dihedral Artin groups A st (see Section 2), which are virtually Z × F for a free group F , see, for example, [6,Lemma 4.3(i)]. The stabilisers of the standard trees of Φ, appearing in (ii), are also Z × F [9, Lemma 4.5] and were described more explicitly in [9,Rm 4.6].…”

Section: Introductionmentioning

confidence: 99%