Tight geodesics in the curve complex

By Brian H. Bowditch. 2008

296 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3ba6d0755fd9798b82ff1af72ddc48d113fc08ac

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In Geometry and Topology

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Citations in the corpus

Stable commutator length on mapping class groups (2013, background): “Bowditch [9] improved this result and showed that the action of MCG(Σ) on C(Σ) is acylindrical.”
Quasi-convexity of hyperbolically embedded subgroups (2013, background): “e [Bow12, Section 7]. This is the case because pseudo-Anosovs act loxodromically on the curve complex [MM99] and the action of a mapping class group on the corresponding curve complex is acylindrical [Bow08] (this is the same reason why ˚is contained in a virtually cyclic hyperbolically embedded sub”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, unknown): “metrically embedded copy of Zfor each UP Bigpgq. Proof of Theorem 7.1. This follows from Lemma 7.3 and Lemma 7.4 below. Our main tool here is the following result of Bowditch: Lemma 7.2 (Lemma 2.2 of [Bow08]). If Gacts acylindrically by isometries on a hyperbolic space M, then each element of Gacts ”
Hyperbolicities in CAT(0) cube complexes (2017, background): “en that, for some speciﬁc spaces, acylindrical and weakly acylindrical actions coincide. For instance, such an equivalence occurs for trees. The ﬁrst non trivial example of this phenomenon appears in [Bow08], in which Bowditch shows that the mapping class group of a surface acts acylindrically on th”
Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group (2008, background): “ lemma compiles three separate results. For case (a), we turn to work of Fujiwara regarding group actions which are “acylindrical,” a property Bowditch deﬁned and proved for the Mod(S)-action on C(S) [Bo]. The statement we need follows from Theorem 3.1(2) in [F]: UNIFORM UNIFORM EXPONENTIAL GROWTH O”
Asymptotic dimension and small‐cancellation for hierarchically hyperbolic spaces and groups (2015, methodology): “The following notion is motivated by work of Bowditch; see [11, Section 3].”
Divergence in lattices in semisimple Lie groups and graphs of groups (2008, background): “s acylindrical if for some l > 0 the stabilizers in Gof pairs of points in Xat distance ≥ lare ﬁnite of uniformly bounded sizes (in this case we say that the action is l-acylindrical). It is known [Bow] that if Xis a tree or a locally ﬁnite hyperbolic graph then for every isometry αof Xsome power”
Divergence, thick groups, and short conjugators (2011, background): “ ﬁnite of uniformly bounded sizes. At times the constant l need not be mentioned. Note that a group acting by isometries on a simplicial tree with unbounded orbits always contains loxodromic elements [Bow08]. Existence of Morse quasi-geodesics implies existence of cut-points in all asymptotic cones.”
Acylindrically hyperbolic groups (2013, methodology): “On the other hand, it is not hard to prove that if a is reducible, then there exists a positive integer n such that [ C G ( a n ) : (cid:12) a n ] = ∞ ; this is impossible if G acts acylindrically on a hyperbolic space and the action of a is loxodromic [8].”
Hyperbolic structures on groups (2017, background): “Recall that an isometric action of a group G on a metric space (S, d) is acylindrical [14] 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, we have #{g ∈ G | d(x, gx) ≤ ε, d(y, gy) ≤ ε} ≤ N.”
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, methodology): “The following lemma is due to Masur-Minsky [91] and Bowditch [28, 29].”
Small cancellation in acylindrically hyperbolic groups (2013, background): “The notion of an acylindrical action was introduced for the special case of groups acting on trees by Sela [34], and in general by Bowditch studying the action of mapping class groups on the curve complex [8].”