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Mating quadratic maps with Kleinian groups via
quasiconformal surgery
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## Mating quadratic maps with Kleinian groups via
quasiconformal surgery

### S. R. Bullett and W. J. Harvey

**Abstract.**
Let $q:\hat{\mathbb C} \to \hat{\mathbb C}$ be any quadratic
polynomial and $r:C_2*C_3 \to PSL(2,{\mathbb C})$ be any faithful
discrete representation of the free product of finite cyclic
groups $C_2$ and $C_3$ (of orders $2$ and $3$) having connected
regular set. We show how the actions of $q$ and $r$ can be
combined, using quasiconformal surgery, to construct a $2:2$
holomorphic correspondence $z \to w$, defined by an algebraic
relation $p(z,w)=0$.

*Copyright 2000 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**06** (2000), pp. 21-30
- Publisher Identifier: S 1079-6762(00)00076-7
- 2000
*Mathematics Subject Classification*. Primary 37F05; Secondary 30D05, 30F40, 37F30
*Key words and phrases*. Holomorphic dynamics, quadratic maps, Kleinian groups, quasiconformal surgery, holomorphic correspondences
- Received by the editors December 22, 1999
- Posted on March 28, 2000
- Communicated by Svetlana Katok
- Comments (When Available)

**S. R. Bullett**

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

*E-mail address:* `s.r.bullett@qmw.ac.uk`

**W. J. Harvey**

Department of Mathematics, King's College, University of London,
Strand, London WC2R 2LS, United Kingdom

*E-mail address:* `bill.harvey@kcl.ac.uk`

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