A combination theorem for covering correspondences and an application to mating polynomial maps with Kleinian groups

Author:
Shaun Bullett

Journal:
Conform. Geom. Dyn. **4** (2000), 75-96

MSC (2000):
Primary 37F05; Secondary 30D05, 30F40

DOI:
https://doi.org/10.1090/S1088-4173-00-00056-4

Published electronically:
April 27, 2000

MathSciNet review:
1755900

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Abstract | References | Similar Articles | Additional Information

The simplest version of the Maskit-Klein combination theorems concerns the action of a free product of two finite subgroups of on the Riemann sphere , when these subgroups have fundamental domains whose interiors together cover . We prove an analogous combination theorem for covering correspondences of rational maps, making use of Douady and Hubbard's Straightening Theorem for polynomial-like maps to describe the structure of the limit sets. We apply our theorem to construct holomorphic correspondences which are matings of polynomial maps with Hecke groups , and we show how it may also be applied to the analysis of separable correspondences.

**1.**S. Bullett and C. Penrose,*Mating quadratic maps with the modular group*, Inventiones Mathematicae**115**(1994), 483-511. MR**95c:58148****2.**S. Bullett and C. Penrose,*A gallery of iterated correspondences*, Experimental Mathematics**3**(1994), 85-105. MR**96a:58156****3.**S. Bullett and C. Penrose,*Regular and limit sets for holomorphic correspondences*, QMW preprint 1999.**4.**S. Bullett and C. Penrose,*Perturbing circle-packing Kleinian groups as correspondences*, Nonlinearity**12**(1999), 635-672. MR**2000d:37054****5.**S. Bullett and W. Harvey,*Mating quadratic maps with Kleinian groups via quasiconformal surgery*, Electron. Res. Announc. Amer. Math. Soc.**6**(2000), 21-30.**6.**A. Douady and J. Hubbard,*On the dynamics of polynomial-like mappings*, Ann. Sci. École Norm. Sup.**18**(1985), 287-343. MR**87f:58083****7.**F. Klein,*Neue Beiträge zur Riemann'schen Functiontheorie*, Math. Ann.**21**(1883), 141-218.**8.**B. Maskit,*On Klein's combination theorem*, Trans. Amer. Math. Soc.**120**(1965), 499-509. MR**33:274****9.**B. Maskit,*On Klein's combination theorem*. II, Trans. Amer. Math. Soc.**131**(1968), 32-39. MR**36:6618****10.**B. Maskit,*On Klein's combination theorem.*III, Advances in the Theory of Riemann Surfaces, Ann. of Math. Studies 66, Princeton University Press, 1971, pp. 297-316. MR**44:6955****11.**B. Maskit,*On Klein's combination theorem.*IV, Trans. Amer. Math. Soc.**336**(1993), 265-294. MR**93e:30088**

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Additional Information

**Shaun Bullett**

Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, University of London, Mile End Road, London E1 4NS, United Kingdom

Email:
s.r.bullett@qmw.ac.uk

DOI:
https://doi.org/10.1090/S1088-4173-00-00056-4

Keywords:
Holomorphic dynamics,
polynomial maps,
Kleinian groups

Received by editor(s):
September 30, 1999

Received by editor(s) in revised form:
January 20, 2000

Published electronically:
April 27, 2000

Additional Notes:
I would like to thank Christopher Penrose for many helpful discussions concerning this work.

Article copyright:
© Copyright 2000
American Mathematical Society