Comparison theorems and orbit counting in hyperbolic geometry

Pollicott, Mark; Sharp, Richard

doi:10.1090/S0002-9947-98-01756-5

Comparison theorems and orbit counting in hyperbolic geometry
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by Mark Pollicott and Richard Sharp

Trans. Amer. Math. Soc. 350 (1998), 473-499

DOI: https://doi.org/10.1090/S0002-9947-98-01756-5

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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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