Browsing by Subject "Anosov representations"

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    A local-to-global principle for Morse quasigeodesics
    (2021-07-21) Riestenberg, John Maxwell; Danciger, Jeffrey; Allcock, Daniel; Bowen, Lewis; Reid, Alan
    In [KLP14], Kapovich, Leeb and Porti gave several new characterizations of Anosov representations Γ → G, including one where geodesics in the word hyperbolic group Γ map to "Morse quasigeodesics" in the associated symmetric space G/K. In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics and describe an algorithm which can verify the Anosov property of a given representation in finite time. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. In this paper, we supplement their work with estimates in the symmetric space to obtain the first explicit criteria for their local-to-global principle. This makes their algorithm for verifying the Anosov property effective. As an application, we demonstrate how to compute explicit perturbation neighborhoods of Anosov representations with two examples.
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    Higher-rank generalizations of convex cocompact and geometrically finite dynamics
    (2022-06-29) Weisman, Theodore Joseph; Danciger, Jeffrey; Allcock, Daniel; Bowen, Lewis; Ballas, Samuel
    We study several higher-rank generalizations of the dynamical behavior of convex cocompact groups in rank-one Lie groups, in the context of both convex projective geometry and relatively hyperbolic groups. Our results include a dynamical characterization of a notion of convex cocompact projective structure due to Danciger-Guéritaud-Kassel. This generalizes a dynamical characterization of Anosov representations of hyperbolic groups. Using topological dynamics, we also define a new notion of geometrical finiteness in higher rank which generalizes previous notions of relative Anosov representation due to Kapovich-Leeb and Zhu. We prove that these “extended geometrically finite” representations are stable under certain small relative deformations, and we provide various examples coming from the theory of convex projective structures.