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Analytic Families of Homomorphisms into PSL(2, ℂ)

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Abstract

We develop a systematic theory of families of homomorphisms

$h_{t}:G\rightarrow {\rm PSL}(2,C),\qquad t\in T,$

that depend analytically on the complex parameter t. Important results have been obtained by Jørgensen, Riley and Sullivan.

The main new concept is the singular set S of parameter values t for which some Moebius transformation h t(x) becomes non-loxodromic. We study the structure and geometry of S and the behaviour of h t in domains \(V \subset T\backslash \bar S\), in particular the extension to values in ∂V ∩ ∂T.

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Authors and Affiliations

  1. Universidad Nacional de Colombia, Escuela de Matemáticas, A. A. 3840 Medellín, Colombia

    Diego Mejía

  2. Institut für Mathematik, Technische Universität Berlin, Fakultät II, MA 8-2, D-10623, Berlin, Germany

    Christian Pommerenke

Authors
  1. Diego Mejía
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  2. Christian Pommerenke
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Corresponding author

Correspondence to Diego Mejía.

Additional information

Research supported by COLCIENCIAS-COLOMBIA, Grant No. 436-2007 and by the project MTM 2006-14449-C02-01 of the Spanish Government and the European Union.

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Mejía, D., Pommerenke, C. Analytic Families of Homomorphisms into PSL(2, ℂ). Comput. Methods Funct. Theory 10, 81–96 (2010). https://doi.org/10.1007/BF03321756

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  • DOI: https://doi.org/10.1007/BF03321756

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