Morse actions of discrete groups on symmetric space
By Michael Kapovich, B. Leeb, J. Porti. 2014
76 citations recorded by the source corpus.
We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex cocompactness. We also prove that our notions are equivalent to the notion of Anosov subgroup, for which we provide a closely related, but simplified and more accessible reformulation, avoiding the geodesic flow of the group. We show moreover that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group. A substantial part of the paper is devoted to the coarse geometry of these discrete subgroups. A key concept which emerges from our analysis is that of Morse quasigeodesics in higher rank symmetric spaces, generalizing the Morse property for quasigeodesics in Gromov hyperbolic spaces. It leads to the notion of Morse actions of word hyperbolic groups on symmetric spaces,i.e. actions for which the orbit maps are Morse quasiisometric embeddings, and thus provides a coarse geometric characterization for the class of subgroups considered in this paper. A basic result is a local-to-global principle for Morse quasigeodesics and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse "Schottky subgroups'' of higher rank semisimple Lie groups via arguments not based on Tits' ping-pong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher rank symmetric spaces.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a559e7dad5fe444b4f3c3a7d5105c7b58fc98689
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Citations in the corpus
- Dynamics on flag manifolds: domains of proper discontinuity and cocompactness (2013, methodology): “We gave the first flow-free definitions in [19].”
- Convex cocompactness in pseudo-Riemannian hyperbolic spaces (2017, unknown): “ on P(Rp,q). We now use a version of the argument of [KLPa, Prop.2.5], inspired by Sullivan’s dynamical characterization [Su] of convex cocompactness in Riemannian hyperbolic spaces. (The argument in [KLPa] is a little more technical because it deals with bundles, whereas we work directly in P(Rp,q)”
- Conformality for a robust class of non-conformal attractors (2019, methodology): “We will not use the original definition but follow a more recent approach, developed by Kapovich-Leeb-Porti [32], Géritaud-Guichard-Kassel-W.”
- A Morse lemma for quasigeodesics in symmetric spaces and euclidean buildings (2014, methodology): “y set Θ. In case (iii), the common tip z can be chosen uniformly close to any point on q. 3 In other words, each coarsely uniformly regular quasigeodesic in X is a Morse quasigeodesic in the sense of [KLP2]. We deduce from this result that coarsely uniformly regularly quasiisometrically embedded sub”
- Finsler bordifications of symmetric and certain locally symmetric spaces (2015, background): “. 7. Γ is S-cocompact. Proof. The implications 1ñ2ñ3are immediate. The equivalence 3ô4 is one of the main results of [KLP3], see Corollary 1.6 of that paper. The equivalences 4ô5ô6 are established in [KLP2]. The implication 5ñ7 is Theorem 10.19 of this paper, while the implication 7ñ1 is established”
- Anosov subgroups: dynamical and geometric characterizations (2017, methodology): “(This is suggested, for instance, by the geometry of free Anosov subgroups, see our examples of Morse– Schottky subgroups in [20] and [16, Section 3.”