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Computational Methods and Function Theory 10 (2010), No. 1, 81--96
Copyright Heldermann Verlag 2010

Analytic Families of Homomorphisms into PSL(2,C)

Diego Mejía
dmejia@unal.edu.co , Universidad Nacional de Colombia, Escuela de Matemáticas, A.A. 3840 Medellín, Colombia.

Christian Pommerenke
pommeren@math.tu-berlin.de , Technische Universität Berlin, Fakultät II, Institut für Mathematik, MA 8-2, D-10623 Berlin, Germany.

[Abstract-pdf] [Abstract-ps]

We develop a systematic theory of families of homomorphisms
ht : G→ PSL(2,C),    t ∈ 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 ht(x) becomes non-loxodromic. We study the structure and geometry of S and the behaviour of ht in domains V⊂ T\setminus\overline{S}, in particular the extension to values in ∂ V∩ ∂ T.

Keywords: Analytic family, homomorphism, Kleinian group, singular set, Moebius transformation, trace, non-elementary group.

MSC 2000: 30F40, 20G20, 20H10.

[FullText-pdf (338 K)] [FullText-ps (565 K)]