The geometry of the curve graph of a right-angled Artin group

For every group G, we define the set of hyperbolic structures on G, denoted H(G), which consists of equivalence classes of (possibly infinite) generating sets of G such that the corresponding Cayley graph is hyperbolic; two generating sets of G are equivalent if the corresponding word metrics on G are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded G-actions on hyperbolic spaces. We are especially interested in the subset AH(G) ⊆ H(G) of acylindrically hyperbolic structures on G, i.e., hyperbolic structures corresponding to acylindrical actions. Elements of H(G) can be ordered in a natural way according to the amount of information they provide about the group G. The main goal of this paper is to initiate the study of the posets H(G) and AH(G) for various groups G. We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of G, and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.We denote the set of hyperbolic structures by H(G) and endow it with the order induced from G(G).Since hyperbolicity of a space is a quasi-isometry invariant, the definition above is independent of the choice of a particular representative in the equivalence class [X]. Using the standard argument from the proof of the Svarc-Milnor Lemma, it is easy to show that elements of H(G) are in one-to-one correspondence with equivalence classes of cobounded actions of G on hyperbolic spaces considered up to a natural equivalence: two actions G S and G T are equivalent if there is a coarsely G-equivariant quasi-isometry S → T .We are especially interested in the subset of acylindrically hyperbolic structures on G, denoted AH(G), which consists of hyperbolic structures [X] ∈ H(G) such that the action of G on the corresponding Cayley graph Γ(G, X) is acylindrical. Recall that an isometric action of a group G on a metric space (S, d) is acylindrical [15] if for every constant ε there exist constants R = R(ε) and N = N (ε) such that for every x, y ∈ S satisfying d(x, y) ≥ R,Groups acting acylindrically on hyperbolic spaces have received a lot of attention in the recent years. For a brief survey we refer to [64].The goal of our paper is to initiate the study of the posets H(G) and AH(G) for various groups G and suggest directions for the future research. Our main results are discussed in the next section. Some open problems are collected in 8.Acknowledgements. The authors would like to thank Spencer Dowdall for useful conversations about mapping class groups, and Henry Wilton, Dani Wise and Hadi Bigdely for helpful discussions of 3-manifold groups. The authors also thank the anonymous referee for useful comments.

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“…For n = 1 both (43) and (44) are obvious. Assume now that (43) and (44) hold for some n ≥ 1; we want to prove the lemma for n + 1 points.…”

Section: Coarsely Isospectral Actionsmentioning

confidence: 99%

Hyperbolic structures on groups

Abbott

Balasubramanya

Osin

2019

Algebr. Geom. Topol.

110

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“…• Υ is connected, triangle-free and square-free without any vertices of degree one (Kim & Koberda, 2014);…”

Section: Introductionmentioning

confidence: 99%

On parabolic subgroups of Artin–Tits groups of spherical type

Cumplido¹,

Gebhardt²,

González-Meneses³

et al. 2019

Advances in Mathematics

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Let A and A be two Artin groups of spherical type, and let A 1 , . . . , A p (resp. A 1 , . . . , A q ) be the irreducible components of A (resp. A ). We show that A and A are commensurable if and only if p = q and, up to permutation of the indices, A i and A i are commensurable for every i. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed n, we give a complete classification of the irreducible Artin groups of rank n that are commensurable with the group of type A n . Note that it will remain 6 pairs of groups to compare to get the complete classification of Artin groups of spherical type up to commensurability. Proposition 2. Let Γ and Ω be two Coxeter graphs of spherical type. If A[Γ] and A[Ω] are commensurable, then Γ and Ω have the same number of vertices. Proof. Suppose that A[Γ] and A[Ω] are commensurable. Let n be the number of vertices of Γ and let m be the number of vertices of Ω. We know that the cohomological dimension of A[Γ] is n and the cohomological dimension of A[Ω] is m (Paris, 2004, Proposition 3.1). As every finite index subgroup of A[Γ] has the same cohomological dimension as A[Γ] and every finite index subgroup of A[Ω] has the same cohomological dimension as A[Ω], we have n = m.

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“…The following lemma allows us to compute the star length for an element in A Γ by using its normal form. Note that this fact can also be derived from Lemma 20 in [KK14].…”

Section: Right-angled Artin Groupsmentioning

confidence: 70%

“…We provide two different algorithms for Theorem B(i). The first algorithm uses the extension graph Γ e [KK13, KK14] and "star metric" on A Γ which is quasi-isometric to Γ e and comparable with the standard normal form. A finitely generated subgroup H of A Γ is stable if and only if its orbit into Γ e with the graph metric is a quasi-isometrically embedding if and only if H is purely loxodromic.…”

Section: Introductionmentioning

confidence: 99%

Algorithms detecting stability and Morseness for finitely generated groups

Kim

2020

Journal of Algebra

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The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group G, Kapovich [K96] provided a partial algorithm which, on input a finite set S of G, halts if S generates a quasiconvex subgroup of G and runs forever otherwise. In this paper, we give various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and finitely generated groups discriminated by a locally quasiconvex torsion-free hyperbolic group (for example, ordinary limit groups).

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The geometry of the curve graph of a right-angled Artin group

Cited by 75 publications

References 28 publications

Hyperbolic structures on groups

Hyperbolic structures on groups

On parabolic subgroups of Artin–Tits groups of spherical type

Algorithms detecting stability and Morseness for finitely generated groups

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