Divergence and quasimorphisms of right-angled Artin groups

By Jason A. Behrstock, Ruth Charney. 2010

137 citations recorded by the source corpus.

We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group AΓ with connected defining graph. We use this to prove that the divergence of AΓ is linear if Γ is a join and quadratic otherwise. As an application, we give a complete description of the cut points in any asymptotic cone of AΓ. We also show that every non-abelian subgroup of AΓ has an infinite-dimensional space of non-trivial quasimorphisms.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d413d4eb9cf2d53f4f3bc9ded26c0c7de274336c

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “15 above to fix the argument from [13].”
Statistically convex-cocompact actions of groups with contracting elements (2016, background): “ulation gives a contradiction since the map n→ hnois a quasi-isometric embedding. Letting n=ni−nj, we obtain ghng−1 =h±n. The description of E(h) thus follows. In what follows, the contracting subset (8) Ax(h)={f⋅o∶f∈E(h)} shall be called the axis of h. Two contracting elements h1;h2 ∈Gare called in”
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014, background): “It was first studied in [24] and [23], and in recent years, for example, in [8, 9, 10, 19, 21, 37, 45].”
Constructing group actions on quasi-trees and applications to mapping class groups (2010, methodology): “ments are WPD although the action of the mapping class group is not properly discontinuous. See [BF09]. There are classiﬁcations of rank 1 elements in Coxeter groups [CF10], right angled Artin groups [BC12] and cube complexes [CS11]. 11 (4) Gis the mapping class group, acting on Teichmu¨ller space w”
The median class and superrigidity of actions on CAT(0) cube complexes (2012, methodology): “The concept of bridge of two parallel hyperplanes was introduced by Behrstock–Charney [3].”
Divergence in right-angled Coxeter groups (2012, background): “Abrams et al [1] and independently Behrstock–Charney [2] have shown that if AΓ is the right-angled Artin group associated to a graph Γ, the group AΓ has either linear or quadratic divergence, and its divergence is linear if and only if Γ is (the 1-skeleton of) a join.”
Contracting boundaries of CAT(0) spaces (2013, background): “These notions have proved fruitful in analyzing right angled Artin groups [3], Teichmüller space [2, 7, 8, 9], the mapping class group [2], CAT(0) spaces [4, 22, 29], and Out(Fn) [1] among others (see also [13, 14, 19, 20]).”
Rank Rigidity for Cat(0) Cube Complexes (2010, background): “It also holds within the class of buildings and Coxeter groups (see [CF10]) as well as for right-angled Artin groups (see [BC10]).”
Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups (2016, methodology): “Strongly separated hyperplanes were introduced in [BC12] in order to characterize the rank-one isometries of right-angled Artin groups; they were also crucial in the proof of the Rank Rigidity Theorem for CAT(0) cube complexes in [CS11].”
Quasiflats in hierarchically hyperbolic spaces (2017, background): “it, we give some intuition coming from CAT(0) cube complexes: Remark. The bridge lemma is well-illustrated by an analogy to CAT(0) cube complexes, where the notion was introduced by Behrstock{Charney [BC11]. In the analogy, let P;Qbe convex subcomplexes of a CAT(0) cube complex M, and let g P;g Q: M”