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