A NOTE ON A PAPER OF SASAKI

摘要

In this paper [10], Sasaki studied the holomorphic slice S of the space of punctured torus groups determined by the trace equation xy = 2z. He found a simply connected domain E contained in S by using his system of inequalities which characterizes some quasifuchsian punctured torus groups (c.f. [9]). Moreover decomposing the boundary of E into 3 pieces partial deriv E = e_1∪ e_2 ∪ e_3 he showed that e_1 ∪ e_2 is contained in S and e_3 (consisting of two points) is in the boundary partial derivS. In this paper we consider the slice S itself more precisely. In this paper we show that S has a structure of the Teichmueller space of once-punctured tori. More precisely it is so called the (rectangular) Earle slice of puncture torus groups. (For the rhombic Earle slice, see [4].) As a corollary of this result, we can show that S is connected and simply connected. Moreover S is a Jordan domain, which is an application of the work of Minsky on the classification of punctured torus groups (c.f. [8] and [5]).