Multiplicities of periodic orbit lengths for non-arithmetic Hecketriangle groups are discussed. It is demonstrated both numerically andanalytically that at least for certain groups the meanmultiplicity of periodic orbits with exactly the same length increasesexponentially with the length. The main ingredient usedis the construction of joint distribution of periodic orbits when groupmatrices are transformed by field isomorphisms. The method can begeneralized to other groups for which traces of group matrices are integersof an algebraic field of finite degree.
