Correction to “Faithful permutation representations of toroidal regular maps”

Abstract

In [1], we claim that we give a complete list of the possible degrees of a faithful transitive permutation representation of the groups of the toroidal regular maps. Indeed, the list given for type {4, 4} is complete; however, recently we were surprised with the existence of exceptional degrees for the map {3, 6}. After, we struggled to find the reason why there were some degrees missing in our classification. The fact is that there is a gap in one proof, having consequences in two of our main theorems. Our goal is to fill in that gap. In what follows, let G be the group of symmetries of a toroidal regular map of type {4, 4} or {3, 6}, and suppose G is represented as a faithful transitive permutation representation group of degree n. Let T be the translation group generated by unitary independent translations of order s defining the groups of {4, 4} s or {3, 6} s for s ∈ {(s, 0), (s, s)} (see [1]). We recall the following results. Lemma 1.1 [1] [Lemma 1.1] If T is transitive, then n = s 2. An immediate consequence of Lemma 1.1 is that if n = s 2 , T is intransitive. Note that T is a normal subgroup of G that is a direct product of two cyclic groups of order s. Consider that α and β are the actions of the generators of T on a block and let K := α, β. In [1], we consider the cases where K is a cyclic group or a direct product, but these cases do not cover all possibilities. In what follows, we complete all the proofs of [1] where this has any effect.

The degrees of toroidal regular proper hypermaps

“…This paper is a sequel to [3] in which faithful transitive permutation representations of the groups of symmetries of toroidal regular maps were determined and rectified in [2]. In the present paper we complete the classification of toroidal regular hypermaps, answering a question made by Gareth Jones in the Bled Conference in Graph Theory 2019, where the results accomplished in [3] were presented.…”

“…To prove that the action on the triple of points is faithful only if δ = 3d, for some divisor d, we can follow an identical proof as the one presented for Theorem 5.3 of [2]. We note that Lemma 2.1 of [2], that establishes the size of T -orbit, can be used here.…”

“…Therefore, two families of toroidal regular maps of type {6, 3} arise: {6, 3} (s,0) and {6, 3} (s,s) , which are factorizations of the infinite Coxeter group [6,3] by (τ 0 τ 1 τ 2 ) 2s and (τ 0 τ 1 τ 0 τ 1 τ 2 ) 2s , respectively. The number of flags of {6, 3} (s,0) is 12s 2 while the number of flags of {6, 3} (s,s) is 36s 2 .…”

“…The degrees of a faithful transitive permutation representation of the group of a regular map of type {3, 6} (or equivalently {6, 3}) are given in [2] by the following two theorems. The number m on Lemma 3.3 is the number of T -orbits, where T is the translation group with the translations as defined in Section 2 of this paper and of [3].…”

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).…”