On Colorings Induced by Low-Index Subgroups of Some Hyperbolic Triangle Group

Abstract

Tilings have been studied in mathematics since ancient times, traditionally within the framework of Euclidean geometry. In recent years, growing interest has emerged in non-Euclidean tilings, particularly those in spherical and hyperbolic geometry, where regular convex polygons generate highly structured and visually striking patterns. Beyond their theoretical importance, hyperbolic tilings offer meaningful applications in areas such as artistic design, architectural pattern formation, mathematical education, and computer-based visualization, where symmetry and color play a central role. This study investigates colorings induced by low-index subgroups of the hyperbolic triangle group *732. Specifically, it constructs right coset colorings of the group, identifies the structural components of its low-index subgroups, and establishes subgroup properties in relation to their induced colorings. Computational tools were employed using GAP (Groups, Algorithms, and Programming) to generate low-index subgroups of the triangle group *732 and to produce the corresponding right coset colorings of the hyperbolic plane. The results show that tiling the entire hyperbolic plane can be achieved by appropriately joining the vertices of a tiling’s fundamental region to form a single, complete rotation. These findings enhance the understanding of symmetry, subgroup structure, and color organization in hyperbolic geometry. Moreover, the resulting colored tilings provide a mathematical foundation for creating complex visual patterns that may be adapted for architectural surfaces, decorative designs, and interactive educational materials. The study demonstrates how abstract group-theoretic concepts can be translated into visually meaningful representations, thereby bridging pure mathematics with applied and creative disciplines. Future research is recommended to explore induced colorings of low-index subgroups of other hyperbolic triangle groups and related hyperbolic structures.