Chiral maps of given hyperbolic type

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