Two Remarks on Fuchsian Groups of Widom Type

Abstract

For groups of Widom type we prove that the direct integral of the spaces of character automorphic forms of weight 1 with the Haar measure on the group of characters is the space of automorphic forms of weight 1 with respect to commutator of the given group. It gives us an explanation why the whole set of ergodic Jacobi matrices with a fixed homogeneous (absolutely continuous) spectrum can be parameterized by the group of characters of the fundamental group of common resolvent set. Besides, it is shown that a Fuchsian group Γ is of Widom type if and only if the conditional entropy of the partition of the unit circle into singleton sets with respect to the partition into Γ-equivalent classes is finite.