1981/01/01 by Akira Yamada · 1 citation
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Graph theory and applications #Topological and Geometric Data Analysis #Mathematics #Constant (computer programming) #Pure mathematics #Computer science
paper · pdf · doi:10.2996/kmj/1138036373
openalex publication_date 1981/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let G be a Fuchsian group operating on the upper half plane H.For and r>0, let Δ(z, r) be the open disc of radius r and center z.Define G zr to be the subgroup of G generated bywhere d(-, •) is the hyperbolic distance induced by the Poincare metric \dz\/lmz.In this paper all references to distance, lines, discs, etc., will be with respect to the hyperbolic geometry unless otherwise stated. Marden [6] proved the following :THEOREM.There is a constant r>0 such that, for any Fuchsian group G and zH, the subgroup G z>r is either cyclic or infinite dihedral (i.e. is generated by two elliptic transformations of order 2).Let μ(z, G) be the supremum of the set of constants r satisfying the conclusion of the Theorem.In fact, this is the maximum by discreteness.Set μ(G)= inf μ(z, G) and μ=inf μ(G).zξΞH G μ will be called Marden's constant in this paper.The purpose of the paper is to determine Marden's constant explicitly.Our result is the following: THEOREM 1.For any Fuchsian group G we have -=0.131467•••with equality occurring precisely when G is the (2, 3, 7) triangle group.If we restrict ourselves to the case where G is torsion-free, then much better bound is obtained.THEOREM 2. For any torsion-free Fuchsian group G we have