On Nonuniqueness of Geodesics in Asymptotic Teichmüller Space

Abstract

In an infinite-dimensional Teichmuller space, it is known that the geodesic connecting two points can be unique or not. In this paper, we study the situation on the geodesic in the universal asymptotic Teichmuller space \(AT(\Delta )\). We introduce the notions of substantial point and non-substantial point in \(AT(\Delta )\). The set of all non-substantial points is open and dense in \(AT(\Delta )\). It is shown that there are infinitely many geodesics joining a non-substantial point to the basepoint. Although we have difficulty in dealing with the substantial points, we give an example to show that there are infinitely many geodesics connecting a certain substantial point and the basepoint. It is also shown that there are always infinitely many straight lines containing two points in \(AT(\Delta )\).