  Asymptotic laws for mean multiplicities of lengths of closed geodesics in
arithmetic hyperbolic three-orbifolds are derived. The sharpest results are
obtained for non-compact orbifolds associated with the Bianchi groups SL(2,o)
and some congruence subgroups. Similar results hold for cocompact arithmetic
quaternion groups, if a conjecture on the number of gaps in their length
spectra is true. The results related to the groups above give asymptotic lower
bounds for the mean multiplicities in length spectra of arbitrary arithmetic
hyperbolic three-orbifolds. The investigation of these multiplicities is
motivated by their sensitive effect on the eigenvalue spectrum of the
Laplace-Beltrami operator on a hyperbolic orbifold, which may be interpreted as
the Hamiltonian of a three-dimensional quantum system being strongly chaotic in
the classical limit.
