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Abstract We prove that infinite orbits of Zariski dense hyperbolic groups equidistribute in homogeneous spaces, in the sense that the family of measures obtained by averaging along spheres in the Cayley graph converges to Haar measure. 

We establish central limit theorems for an action of a group $$G$$ on a hyperbolic space $$X$$ with respect to the counting measure on a Cayley graph of $$G$$ . Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds. 

Abstract For a pseudo-Anosov flow $$\varphi $$ without perfect fits on a closed $$3$$ -manifold, Agol–Guéritaud produce a veering triangulation $$\tau $$ on the manifold M obtained by deleting the singular orbits of $$\varphi $$ . We show that $$\tau $$ can be realized in M so that its 2-skeleton is positively transverse to $$\varphi $$ , and that the combinatorially defined flow graph $$\Phi $$ embedded in M uniformly codes the orbits of $$\varphi $$ in a precise sense. Together with these facts, we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of the closed orbits of $$\varphi $$ after cutting M along certain transverse surfaces, thereby generalizing the work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M . Our work can be used to study the flow $$\varphi $$ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the ‘positive’ cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $$3$$ -manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.