REFLEXIBLE REGULAR MAPS WITH NO NON-TRIVIAL EXPONENTS FROM RESIDUAL FINITENESS

Abstract

AbstractRegular maps, that is, graph embeddings with the ‘highest level’ of orientation-preserving symmetry, can be identified with two-generator presentations of groups G of the form 〈x, y; xm = y2 = (xy)n = … = 1〉; the parameters m and n are the valence and the covalence of the map, respectively. The element j ∈ Zm* is an exponent of such a map if the assignments x ↦ xj and y ↦ y extend to an automorphism of G. The element −1 is an exponent if and only if the map is reflexible, that is, isomorphic to its mirror image. Non-trivial exponents, j ≠ ±1, induce automorphisms of the underlying graph, which are not map automorphisms but can nevertheless be considered as ‘external symmetries’ of the map. In this paper we show with the help of residual finiteness of triangle groups that for any given m, n≥3 such that 1/m + 1/n ≤ 1/2, there exist infinitely many finite, reflexible and regular maps of valence m and covalence n with no non-trivial exponent.