Quasi-convexity of hyperbolically embedded subgroups

By Alessandro Sisto. 2013

82 citations recorded by the source corpus.

We show that any infinite order element g of a virtually cyclic hyperbolically embedded subgroup of a group G is Morse, that is to say any quasi-geodesic connecting points in the cyclic group C generated by g stays close to C. This answers a question of Dahmani–Guirardel–Osin. What is more, we show that hyperbolically embedded subgroups are quasi-convex. Finally, we give a definition of what it means for a collection of subspaces of a metric space to be hyperbolically embedded and we show that axes of pseudo-Anosovs are hyperbolically embedded in Teichmüller space endowed with the Weil-Petersson metric.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b95a3d2718eabfc7a39baf0f3a89dbedbc51903c

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Growth tight actions (2014, methodology): “We define path systems and contracting elements following Sisto [52].”
Morse Boundaries of Proper Geodesic Metric Spaces (2015, background): “y respectively. The generality in which this boundary is deﬁned means it is a new quasiisometry invariant for every ﬁnitely generated group. While the Morse boundary of a group may be empty, Sisto in [25] showed that every group in the class of acylindrically hyperbolic groups uniﬁed by Osin in [20]”
On strongly quasiconvex subgroups (2017, background): “ic peripheral subgroups of a relatively hyperbolic group and non-hyperbolic hyperbolically embedded subgroups of a ﬁnitely generated group are non-stable strongly quasiconvex subgroups(see [DS05] and [Sis16]). In Section 7, we also provide many examples of non-stable strongly quasiconvex subgroups i”
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014, background): “Acylindrical hyperbolicity has strong consequences: Every acylindrically hyperbolic group is SQ-universal, it contains free normal subgroups [DGO11], it contains Morse elements and hence all its asymptotic cones have cut-points [Sis16], and its bounded cohomology is infinite dimensional in degrees 2”
Hyperbolicities in CAT(0) cube complexes (2017, unknown): “If G is acylindrically hyperbolic, then G contains a Morse element g ∈ G according to [Sis16].”
Acylindrically hyperbolic groups (2013, background): “Theorem 8.8 (Sisto [49]) .”