Every group is the automorphism group of a graph with arbitrarily large genus

TL;DR

This paper demonstrates that for any abstract group, there exists a sequence of graphs whose automorphism groups are isomorphic to that group, with the graphs' genus growing arbitrarily large, linking group theory and graph topology.

Contribution

It establishes that every group can be realized as the automorphism group of graphs with unbounded genus, extending the understanding of symmetry groups in graph theory.

Findings

Constructed graphs with prescribed automorphism groups

Proved the genus of these graphs can be arbitrarily large

Connected group theory with topological graph properties

Abstract

We prove that, to every abstract group $G$, we can associate a sequence of graphs $\Gamma_n$ such that the automorphism group of $\Gamma_n$ is isomorphic to $G$ and the genus of $\Gamma_n$ is an unbounded function of $n$.