Classification and Galois conjugacy of Hamming maps

Abstract

We show that for each d ≥ 1 the d -dimensional Hamming graph H ( d , q ) has an orientably regular surface embedding if and only if q is a prime power p e . If q > 2 there are up to isomorphism φ ( q − 1)/ e such maps, all constructed as Cayley maps for a d -dimensional vector space over the field F q . We show that for each such pair ( d , q ) the corresponding Belyi pairs are conjugate under the action of the absolute Galois group Gal \overline Q , and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for q > 3.