On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

Abstract

Let AZ(R) be the infinitesimal asymptotic Teichmüller space of a Riemann surface R of infinite type. It is known that AZ(R) is the quotient Banach space of the infinitesimal Teichmüller space Z(R), where Z(R) is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points in AZ(R). We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller space AZ(D) by constructing a special degenerating sequence.