Tohoku Mathematical Journal

Splitting density for lifting about discrete groups

Yasufumi Hashimoto and Masato Wakayama

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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.

Article information

Tohoku Math. J. (2) Volume 59, Number 4 (2007), 527-545.

First available in Project Euclid: 6 January 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11F72: Spectral theory; Selberg trace formula

Prime geodesic theorem splitting density Selberg's zeta function regular cover congruence subgroup semisimple Lie groups


Hashimoto, Yasufumi; Wakayama, Masato. Splitting density for lifting about discrete groups. Tohoku Math. J. (2) 59 (2007), no. 4, 527--545. doi:10.2748/tmj/1199649873.

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