Non-orientable regular maps with negative prime-power Euler characteristic

2025/07/04 by Conder, Marston, Gill, Nick, Širáň, Jozef · 1 citation
#05E18 #20B25 #20F05 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2507.03667

Abstract

In this paper we provide a classification of all regular maps on surfaces of Euler characteristic -r^d for some odd prime r and integer d≥ 1. Such maps are necessarily non-orientable, and the cases where d = 1 or 2 have been dealt with previously. This classification splits naturally into three parts, based on the nature of the automorphism group G of the map, and particularly the structure of its quotient G/O(G) where O(G) is the largest normal subgroup of G of odd order. In fact G/O(G) is isomorphic to either a 2-group (in which case G is soluble), or \textrmPSL(2,q) or \textrmPGL(2,q) where q is an odd prime power. The result is a collection of 18 non-empty families of regular maps, with conditions on the associated parameters.

Citations

• Automorphism groups of Riemann surfaces of genus p+1, where p is prime

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• On orientably-regular maps of Euler characteristic -2p²

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