# On large groups of symmetries of finite graphs embedded in spheres

## TL;DR

This paper investigates the symmetry groups of finite graphs embedded in spheres across various dimensions, establishing bounds on the group order relative to the graph's genus and exploring minimal embedding dimensions.

## Contribution

It generalizes previous 3D results to arbitrary dimensions, providing polynomial bounds on symmetry group orders and analyzing minimal embedding dimensions.

## Key findings

- Order of symmetry group bounded by polynomial in genus
- Degree of polynomial depends on sphere dimension
- Optimal bounds established for even dimensions

## Abstract

Let G be a finite group acting orthogonally on a pair (S^d,\Gamma) where \Gamma is a finite, connected graph of genus g>1 embedded in the sphere S^d. The 3-dimensional case d=3 has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus g>1 the maximum order of a G-action on a pair (S^3,\Gamma) is determined and the corresponding graphs \Gamma are classified. In the present paper we consider arbitrary dimensions d and prove that the order of G is bounded above by a polynomial of degree d/2 in g if d is even, and of degree (d+1)/2 if d is odd; moreover the degree d/2 is best possible in even dimensions d. We discuss also the problem, given a finite graph \Gamma and its finite symmetry group, to find the minimal dimension of a sphere into which \Gamma embeds equivariantly as above.