Tohoku Mathematical Journal
2007
December
SECOND SERIES VOL. 59, NO. 4

Tohoku Math. J.
59 (2007), 527-545

Title SPLITTING DENSITY FOR LIFTING ABOUT DISCRETE GROUPS

Dedicated to Ryoshi Hotta on the occasion of his sixty-fifth birthday
Author Yasufumi Hashimoto and Masato Wakayama

(Received March 2, 2006, revised June 5, 2007)
Abstract. We study splitting densities of primitive elements of a discrete subgroup of a connected non-compact semisimple Lie group of real rank one with finite center in another larger such discrete subgroup. When the corresponding cover of such a locally symmetric negatively curved Riemannian manifold is regular, the densities can be easily obtained from the results due to Sarnak or Sunada. Our main interest is a case where the covering is not necessarily regular. Specifically, for the case of the modular group and its congruence subgroups, we determine the splitting densities explicitly. As an application, we study analytic properties of the zeta function defined by the Euler product over elements consisting of all primitive elements which satisfy a certain splitting law for a given lifting.

2000 Mathematics Subject Classification. Primary 11M36; Secondary 11F72.
Key words and phrases. Prime geodesic theorem, splitting density, Selberg's zeta function, regular cover, congruence subgroup, semisimple Lie groups.

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