On a Canonical Construction of Tessellated Surfaces From Finite Groups

Explore a canonical construction that associates tessellated surfaces with finite groups in this 59-minute lecture by Jon Pakianathan. Delve into an elementary construction that links the non-commutative part of a finite group's multiplication table to a finite collection of closed, connected, oriented surfaces with specific cell structures. Discover how these structures, known as "dual quasiregular," feature face and edge-transitive properties, with all faces on a particular surface being n-gons for fixed n and at most two different vertex valences. Examine examples ranging from regular cell structures like platonic solids and their higher genus generalizations to quasi-regular structures with varying vertex valences. Learn about the group's conjugation action on these surfaces, inducing faithful, orientation-preserving actions of subquotients. Understand how these surfaces form 3-fold branched covers over the Riemann sphere, inheriting unique complex structures compatible with the group action. Time permitting, explore connections between this construction and classical concepts such as Coxeter, Klein, triangle and Fuchsian groups, graph embedding, and Grothendieck's dessin d'enfant construction. Gain insights from numerous examples discussed throughout the lecture, based on joint work with Mark Herman published in Topology and its Applications.