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The exponent of convergence of Poincaré series of combination groups. (English) Zbl 0739.20007

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.

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)
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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
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