The degrees of toroidal regular proper hypermaps

Abstract

Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished. In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type (3, 3, 3).

Proper locally spherical hypertopes of hyperbolic type

“…Recently Pellicer, Toledo and Potočnik used GPR-graphs to build 2-orbit maniplexes for every rank and every symmetry type. In another direction in [15] Fernandes and Piedade classify the CPR-graphs of regular maps on tori and in [16] they extend those results to regular toroidal hypermaps.…”

The Degrees of Regular Polytopes of Type [4, 4, 4]

“…In [9] we gave the list of all possible degrees for toroidal regular maps (for the regular toroidal map of type {3, 6} the degrees given in [9] were rectified in [10]). In [11] we completed the investigation on a surface of genus 1, and rank 3, considering the group of a regular toroidal hypermap of type (3,3,3).…”