We show that the Poincaré series of the Fuchsian group of deck transformations of ℂ \ ℤ diverges logarithmically. This is because ℂ \ ℤ is a ℤ-cover of the three horned sphere, whence its geodesic flow has a good section which behaves like a random walk on ℝ with Cauchy distributed jump distribution and has logarithmic asymptotic type.
|Number of pages||20|
|Journal||Ergodic Theory and Dynamical Systems|
|State||Published - Feb 1999|