On the Cartwright-Steger surface

Abstract

In this article, we study various concrete algebraic and differential geometric properties of the Cartwright-Steger surface. In particular, we determine the genus of a generic fiber of the Albanese fibration, and deduce that the singular fibers are not totally geodesic, answering an open problem about fibrations of a complex ball quotient over a Riemann surface.

A longer version of this paper, which contains some geometric results we have omitted here, provides more details of some calculations, and adopts a different approach to proving some results, can be found on the webpage of the first named author (see reference [CKY]). It is an update of version 1 of this paper