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Proceedings of the American Mathematical Society

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The geodesic flow for discrete groups of infinite volume

Author: Peter J. Nicholls
Journal: Proc. Amer. Math. Soc. 96 (1986), 311-317
MSC: Primary 58F17
MathSciNet review: 818464
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Abstract: Let $ \Gamma $ be a discrete group acting in the unit ball $ B$ of euclidean $ n$-space and $ T(B)$ the unit tangent space of $ B$. We define the geodesic flow $ {g_t}$ on the quotient space $ \Omega = T(B)/\Gamma $ and show that for discrete groups of infinite volume the flow is of zero type--namely, for measurable subsets $ A,B$ of $ \Omega $ which are of finite measure, $ {\lim _{t \to \infty }}{g_t}(A) \cap B = 0$. Using this result, we give a new and elementary proof of the fact that for a discrete group of infinite volume, $ N(r) = o(V\{ x:\left\vert x \right\vert < r\} )$ as $ r \to 1$, where $ N(r)$ is the orbital counting function and $ V$ denotes hyperbolic volume.

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Article copyright: © Copyright 1986 American Mathematical Society

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