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Free, publicly-accessible full text available August 6, 2025
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This is the second in a series of two papers that develops a theory of relatively Anosov representations using the original “contracting flow on a bundle” definition of Anosov representations introduced by Labourie and Guichard–Wienhard. In this paper, we focus on building families of examples.more » « lessFree, publicly-accessible full text available June 1, 2025
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Forni, Giovanni (Ed.)We study Patterson–Sullivan measures for a class of discrete subgroups of higher rank semisimple Lie groups, called transverse groups, whose limit set is well-defined and transverse in a partial flag variety. This class of groups includes both Anosov and relatively Anosov groups, as well as all discrete subgroups of rank one Lie groups. We prove an analogue of the Hopf–Tsuji–Sullivan dichotomy and then use this dichotomy to prove a variant of Burger's Manhattan Curve Theorem. We also use the Patterson–Sullivan measures to obtain conditions for when a subgroup has critical exponent strictly less than the original transverse group. These gap results are new even for Anosov groups.more » « lessFree, publicly-accessible full text available January 1, 2025
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Abstract A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.
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Free, publicly-accessible full text available November 15, 2024
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In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic domain has certain visibility properties, with a focus on unbounded domains. “Visibility” in this context is reminiscent of visibility, seen in negatively curved Riemannian manifolds, in the sense of Eberlein–O’Neill. However, we do not assume that the domains studied are Cauchy-complete with respect to the Kobayashi distance, as this is hard to establish for domains in C d \mathbb {C}^d , d ≥ 2 d \geq 2 . We study the various ways in which this property controls the boundary behavior of holomorphic maps. Among these results is a Carathéodory-type extension theorem for biholomorphisms between planar domains—notably: between infinitely-connected domains. We also explore connections between our visibility property and Gromov hyperbolicity of the Kobayashi distance.more » « less