The core chain of circles of Maskit’s embedding for once-punctured torus groups

Abstract:

In this paper, we describe the limit set $\Lambda _n$ of a sequence of manifolds $N_n$ in the boundary of Maskit’s embedding of the once-punctured torus. We prove that $\Lambda _n$ contains a chain of tangent circles $\{C_{n,j}\}$ that are described from the end invariants of the manifold. In particular, we give estimates in terms of $n$ of the radii $r_{n,j}$ of the circles and prove that $r_{n,j}$ decrease when $n$ tends to infinity. We then apply these results to McShane’s identity, to obtain an estimate of the width of the limit set in terms of $n$.