Chiral maps of given hyperbolic type with alternating automorphism group

Abstract This paper proves the existence of a chiral map with alternating automorphism group for every hyperbolic type. Equivalently, for every pair of natural numbers such that , there is a finite alternating group generated by a pair of elements whose orders are and and whose product is an involution, where furthermore the group does not have an automorphism which inverts these generators. We call on previously known results for when both the valency and the face‐length are odd, and present a set of new constructions using permutations for when at least one parameter is even.