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Comparison theorems and orbit counting in hyperbolic geometry

Authors: Mark Pollicott and Richard Sharp
Journal: Trans. Amer. Math. Soc. 350 (1998), 473-499
MSC (1991): Primary 20F32, 22E40, 58E40; Secondary 11F72, 20F10, 58F20
MathSciNet review: 1376553
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Abstract: In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both ``thermodynamic'' ergodic theory and the automaton associated to strongly Markov groups.

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Additional Information

Mark Pollicott
Affiliation: Department of Mathematics, University of Warwick, Coventry, CV4 7AL, U.K.
Address at time of publication: Department of Mathematics, University of Manchester, Oxford Road, Man- chester, M13 9PL, U.K.

Richard Sharp
Affiliation: Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, U.K.
Address at time of publication: Department of Mathematics, University of Manchester, Oxford Road, Man- chester, M13 9PL, U.K.

Keywords: Strongly Markov, hyperbolic group, Kleinian group, orbit counting function, negative curvature, Poincar\'{e} series
Received by editor(s): May 23, 1995
Additional Notes: The first author was supported by The Royal Society through a University Research Fellowship. The second author was supported by the UK SERC under grant number GR/G51930 held at Queen Mary and Westfield College.
Article copyright: © Copyright 1998 American Mathematical Society

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