[Accouplements, correspondances holomorphes, et une tranche de Bers]
There are two frameworks for mating Kleinian groups with rational maps on the Riemann sphere: the algebraic correspondence framework due to Bullett-Penrose-Lomonaco [BP94, BL20] and the simultaneous uniformization mating framework of [MM23a]. The current paper unifies and generalizes these two frameworks. To achieve this, we extend the mating framework of [MM23a] to genus zero hyperbolic orbifolds with at most one orbifold point of order $\nu \ge 3$ and at most one orbifold point of order two. We give an explicit description of the resulting conformal matings in terms of uniformizing rational maps. Using these rational maps, we construct correspondences that are matings of such hyperbolic orbifold groups (including punctured spheres and Hecke groups) with polynomials in real-symmetric hyperbolic components. We also define an algebraic parameter space of correspondences and construct an analog of a Bers slice of the above orbifolds in this parameter space.
Il existe deux cadres pour l’accouplement de groupes kleiniens avec des applications rationnelles sur la sphère de Riemann : le cadre de correspondance algébrique dû à Bullett-Penrose-Lomonaco [BP94, BL20] et le cadre d’accouplement par uniformisation simultanée de [MM23a]. Le présent article unifie et généralise ces deux cadres. Pour ce faire, nous étendons le cadre d’accouplement de [MM23a] aux orbifolds hyperboliques de genre zéro avec au plus un point d’orbifold d’ordre $\nu \ge 3$ et au plus un point d’orbifold d’ordre $2$. Nous donnons une description explicite des accouplements conformes qui en résultent en termes de fonctions rationnelles uniformisantes. À l’aide de ces fonctions rationnelles, nous construisons des correspondances qui sont des accouplements de tels groupes d’orbifolds hyperboliques (y compris les sphères perforées et les groupes de Hecke) avec des polynômes en composantes hyperboliques à symétrie réelle. Nous définissons également un espace de paramètres algébriques de correspondances et construisons un analogue d’une tranche de Bers des orbifolds ci-dessus dans cet espace de paramètres.
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Keywords: Fuchsian group, Bowen-Series map, rational map, simultaneous uniformization, conformal mating, algebraic correspondence, Teichmüller space, Bers slice
Mots-clés : Groupe fuchsien, application de Bowen-Series, fonction rationnelle, uniformisation simultanée, accouplement conforme, correspondance algébrique, espace de Teichmüller, tranche de Bers
Mahan Mj 1 ; Sabyasachi Mukherjee 1
CC-BY 4.0
@article{JEP_2025__12__1445_0,
author = {Mahan Mj and Sabyasachi Mukherjee},
title = {Matings, holomorphic correspondences, and {a~Bers~slice}},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1445--1502},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.315},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.315/}
}
TY - JOUR AU - Mahan Mj AU - Sabyasachi Mukherjee TI - Matings, holomorphic correspondences, and a Bers slice JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 1445 EP - 1502 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.315/ DO - 10.5802/jep.315 LA - en ID - JEP_2025__12__1445_0 ER -
%0 Journal Article %A Mahan Mj %A Sabyasachi Mukherjee %T Matings, holomorphic correspondences, and a Bers slice %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 1445-1502 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.315/ %R 10.5802/jep.315 %G en %F JEP_2025__12__1445_0
Mahan Mj; Sabyasachi Mukherjee. Matings, holomorphic correspondences, and a Bers slice. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1445-1502. doi: 10.5802/jep.315
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