Abstract
In this Independent Study, we explore finite tilings of hyperbolic space as defined on a regular lattice. We adapt a group theoretical invariant developed for this type of problems in Z2 by John Conway and Jeffrey Lagarias. For this, we first present the properties of hyperbolic geometry in the Poincaré Disc Model, its groups of isometries, and Fuchsian groups. Following this, we summarize Conway and Lagarias’ argument in Euclidean space along with their characterization of tilings via boundary words derived from F2 . We then translate their constructions to hyperbolic space through the use of the Poincaré Polygon Theorem together with the resulting regular tesselations of hyperbolic space and their respective Fuchsian groups. Finally, we generalize the Conway-Lagarias invariant while also defining our own tiling problem in the hyperbolic lattice generated by 8 hyperbolic octagons.
Advisor
Kelvey, Robert
Department
Mathematics
Recommended Citation
Jarrín Acosta, Felipe, "A Group Theoretical Approach to Tilings of Regular Hyperbolic Lattices" (2024). Senior Independent Study Theses. Paper 10930.
https://openworks.wooster.edu/independentstudy/10930
Disciplines
Algebra | Discrete Mathematics and Combinatorics | Geometry and Topology
Keywords
Hyperbolic geometry, polyomino, tiling, tesselation, boundary word, Fuchsian group, lattice.
Publication Date
2024
Degree Granted
Bachelor of Arts
Document Type
Senior Independent Study Thesis
© Copyright 2024 Felipe Jarrín Acosta
