On large groups of symmetries of finite graphs embedded in spheres

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.