Chiral maps of given hyperbolic type

Conder, Marston D. E.; Hucíková, Veronika; Nedela, Roman and Širáň, Jozef (2016). Chiral maps of given hyperbolic type. Bulletin of the London Mathematical Society, 48(1) pp. 38–52.

DOI: https://doi.org/10.1112/blms/bdv086

Abstract

We prove the existence of infinitely many orientably-regular but chiral maps of every given hyperbolic type {m, k}, by constructing base examples from suitable permutation representations of the ordinary (2, k, m) triangle group, and then taking abelian p-covers. The base examples also help to prove that for every pair (k, m) of integers with 1/k + 1/m ≤ 1/2, there exist infinitely many regular and infinitely many orientably-regular but chiral maps of type {m, k}, each with the property that both the map and its dual have simple underlying graph.