On large groups of symmetries of finite graphs embedded in spheres

Abstract

Let [Formula: see text] be a finite group acting orthogonally on a pair [Formula: see text] where [Formula: see text] is a finite, connected graph of genus [Formula: see text] embedded in the sphere [Formula: see text]. The 3-dimensional case [Formula: see text] has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus [Formula: see text], the maximum order of an orientation-preserving [Formula: see text]-action on a pair [Formula: see text] is determined and the corresponding graphs [Formula: see text] are classified (an upper bound for the order of [Formula: see text] is [Formula: see text]). In the present paper, we consider arbitrary dimensions [Formula: see text] and prove that the order of [Formula: see text] is bounded above by a polynomial of degree [Formula: see text] in [Formula: see text] if [Formula: see text] is even and of degree [Formula: see text] if [Formula: see text] is odd; moreover, the degree [Formula: see text] is best possible in even dimensions [Formula: see text]. We discuss also the problem, given a finite graph [Formula: see text] and its finite symmetry group, to find the minimal dimension of a sphere into which [Formula: see text] embeds equivariantly as above.