Furusawa, Harushi The exponent of convergence of Poincaré series of combination groups. (English) Zbl 0739.20007 Tôhoku Math. J., II. Ser. 43, No. 1, 1-7 (1991). The author considers the free product of two discrete subgroups of the Möbius group of transformations \(GM(B^{n+1})\) with an amalgamated subgroup. For this general case he gives an estimate from below for the exponent of convergence in terms of the individual exponents. Reviewer: N.V.Kuznetsov (Khabarovsk) Cited in 4 Documents MSC: 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20H05 Unimodular groups, congruence subgroups (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) Keywords:Poincaré series; free product; Möbius group of transformations; exponent of convergence PDFBibTeX XMLCite \textit{H. Furusawa}, Tôhoku Math. J. (2) 43, No. 1, 1--7 (1991; Zbl 0739.20007) Full Text: DOI References: [1] L V. AHLFORS, Mbius Transformationsin Several Dimensions, Univ of Minnesota Lecture Notes, Minnesota, 1981. [2] T. AKAZA, Local property of the singular sets of some Kleinian groups, Thoku Math. J. 25 (1973), 1-22. · Zbl 0264.30025 · doi:10.2748/tmj/1178241411 [3] T. AKAZA AND T. SHIMAZAKI, The Hausdorf dimension of the singular sets of combination groups, Thoku, Math. J. 25 (1973), 61-68. · Zbl 0264.30026 · doi:10.2748/tmj/1178241415 [4] A F BEARDON, The Geometry of Discrete Groups, Springer Verlarg, New York-Heidelberg-Berlin, 1983 · Zbl 0528.30001 [5] B MASKIT, Kleinian Groups, Springer Verlarg, NewYork-Heidelberg-Berlin, 198 · Zbl 0627.30039 [6] S J PATTERSON, The exponent of convergence of Poincareseries, Monatsh F Math 82(1976), 297-31 · Zbl 0349.30012 · doi:10.1007/BF01540601 [7] S J PATTERSON, Lectures on measures on limit sets of Kleinian groups, in Analytical and Geometri Aspects of Hyperbolic Space (D B. Epstein, ed), London Math. Soc. Lecture Notes 111(1984), 281-323 · Zbl 0611.30036 [8] N J WIELENBERG, Discrete Mobius groups: fundamentalpolyhedra and convergence, Amer J. Math 99 (1977), 861-877. JSTOR: · Zbl 0373.57024 · doi:10.2307/2373869 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.