## Mating quadratic maps with Kleinian groups via quasiconformal surgery.(English)Zbl 1027.37024

Summary: Let $$q:\hat{\mathbb C} \to \hat{\mathbb C}$$ be any quadratic polynomial and $$r:C_2*C_3 \to \text{PSL}(2,{\mathbb C})$$ be any faithful discrete representation of the free product of finite cyclic groups $$C_2$$ and $$C_3$$ (of orders $$2$$ and $$3$$) having a 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 $$p(z,w)=0$$.

### MSC:

 37F05 Dynamical systems involving relations and correspondences in one complex variable 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)

### Keywords:

holomorphic dynamics
Full Text:

### References:

 [1] Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385 – 404. · Zbl 0104.29902 [2] Lars V. Ahlfors, Lectures on quasiconformal mappings, Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. · Zbl 0138.06002 [3] Shaun Bullett and Christopher Penrose, Mating quadratic maps with the modular group, Invent. Math. 115 (1994), no. 3, 483 – 511. · Zbl 0801.30025 [4] Shaun Bullett and Christopher Penrose, Perturbing circle-packing Kleinian groups as correspondences, Nonlinearity 12 (1999), no. 3, 635 – 672. · Zbl 0984.37048 [5] S. Bullett, A combination theorem for covering correspondences and an application to mating polynomial maps with Kleinian groups, QMW preprint 1999. [6] W. Chow, On compact complex analytic varieties, Am. J. Math. 71 (1949), 893-914. · Zbl 0041.48302 [7] Adrien Douady and John Hamal Hubbard, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 3, 123 – 126 (French, with English summary). · Zbl 0483.30014 [8] Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287 – 343. · Zbl 0587.30028 [9] W. J. Harvey, Spaces of discrete groups, Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975), Academic Press, London, 1977, pp. 295 – 348. [10] Bernard Maskit, On Klein’s combination theorem, Trans. Amer. Math. Soc. 120 (1965), 499 – 509. · Zbl 0138.06803 [11] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6 (1956), 1-42. · Zbl 0075.30401 [12] Lei Tan, Matings of quadratic polynomials, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 589 – 620. · Zbl 0756.58024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.