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The core chain of circles of Maskit’s embedding for once-punctured torus groups. (English) Zbl 1185.30044

Summary: 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 \( n\).

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M50 General geometric structures on low-dimensional manifolds
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