  Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle
groups are discussed. It is demonstrated both numerically and analytically that
at least for certain groups the mean multiplicity of periodic orbits with
exactly the same length increases exponentially with the length. The main
ingredient used is the construction of joint distribution of periodic orbits
when group matrices are transformed by field isomorphisms. The method can be
generalized to other groups for which traces of group matrices are integers of
an algebraic field of finite degree.
