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{1199649873,
author = {Hashimoto, Yasufumi and Wakayama, Masato},
title = {Splitting density for lifting about discrete groups},
journal = {Tohoku Math. J. (2)},
volume = {59},
number = {1},
year = {2007},
pages = { 527-545},
language = {en},
url = {http://dml.mathdoc.fr/item/1199649873}
}
Hashimoto, Yasufumi; Wakayama, Masato. Splitting density for lifting about discrete groups. Tohoku Math. J. (2), Tome 59 (2007) no. 1, pp. 527-545. http://gdmltest.u-ga.fr/item/1199649873/