Expositio paper exp-20260628-a87bee
Group Actions and Möbius Symmetries in Hyperbolic Tessellations
This expository paper explains tessellations of the hyperbolic plane from a differential-geometric viewpoint. We review the upper half-plane (H) and Poincaré disk (D) models and the Cayley transform linking them. We prove that orientation-preserving isometrie…
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Abstract
This expository paper explains tessellations of the hyperbolic plane from a differential-geometric viewpoint. We review the upper half-plane (H) and Poincaré disk (D) models and the Cayley transform linking them. We prove that orientation-preserving isometries are precisely PSL(2, R) acting by Möbius/Blaschke maps and show these maps are conformal and hyperbolic isometries (hence length-and area-preserving). Using group actions of discrete subgroups (Fuchsian groups), we construct tessellations from fundamental domains; the modular group serves as a guiding example. Via Gauß-Bonnet and reflection/triangle-group constructions, we derive the existence criterion for regular {p, q} tilings, namely 1 p + 1 q < 1/2, and relate angle defect to area. Throughout the article, we highlight geodesics, curvature K ≡ −1, and the role of isometries in organizing hyperbolic tilings.
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