Realization of Regular Maps of Large Genus

Abstract

Regular maps are an algebraic concept to describe most symmetric tilings of closed surfaces of arbitrary genus. All such regular maps resp. symmetric tilings of surfaces up to genus 302 are algebraically known in the form of symmetry groups acting on their universal covering space. But still little is known about geometric realizations, i.e. about finding most symmetric embeddings of closed surfaces with high genus and a supported most symmetric tiling. In this report, we will construct some new highly symmetric embeddings of regular maps of up to genus 61 and thereby shed some new light on this fundamental problem at the interface of algebra, differential geometry, and topology.