Surfaces with given Automorphism Group

TL;DR

This paper constructs surfaces with prescribed automorphism groups, improving existing methods, proving cycle double covers, and embedding these surfaces into 3D space with symmetric properties.

Contribution

It simplifies Frucht's graph construction for two-generator groups, addresses an oversight for general groups, and provides explicit embeddings into Euclidean space.

Findings

Constructed minimal graphs for two-generator groups

Proved existence of cycle double covers for these graphs

Embedded surfaces into 3D space with matching automorphism and symmetry groups

Abstract

Frucht showed that, for any finite group $G$, there exists a cubic graph such that its automorphism group is isomorphic to $G$. For groups generated by two elements we simplify his construction to a graph with fewer nodes. In the general case, we address an oversight in Frucht's construction. We prove the existence of cycle double covers of the resulting graphs, leading to simplicial surfaces with given automorphism group. For almost all finite non-abelian simple groups we give alternative constructions based on graphic regular representations. In the general cases $C_n,D_n,A_5$ for $n\geq 4$, we provide alternative constructions of simplicial spheres. Furthermore, we embed these surfaces into the Euclidean 3-Space with equilateral triangles such that the automorphism group of the surface and the symmetry group of the corresponding polyhedron in $\mathrm{O}(3)$ are isomorphic.