Chirality groups of maps and hypermaps

Abstract

Abstract. Although the phenomenon of chirality appears in many investigations of maps and hypermaps no detailed study of chirality seems to have been carried out. Chirality of maps and hypermaps is not merely a binary invariant but can be quantified by two new invariants -the chirality group and the chirality index, the latter being the size of the chirality group. A detailed investigation of the chirality groups of maps and hypermaps will be the main objective of this paper. The most extreme type of chirality arises when the chirality group coincides with the monodromy group. Such hypermaps are called totally chiral. Examples of them are constructed by considering appropriate "asymmetric" pairs of generators for some non-abelian simple groups. We also show that every finite abelian group is the chirality group of some hypermap, whereas many non-abelian groups, including symmetric and dihedral groups, cannot arise as chirality groups.

Rank 4 toroidal hypertopes

“…The existence of these objects in any rank was proved in [12]. There is also a notion of chirality in hypermaps as well, see for example, [2]. Similarly we say for a hypertope to be chiral if its automorphism group action has two chamber orbits and adjacent chambers are in different orbits [7].…”

Constructions of Chiral Polytopes of Small Rank

“…We now describe the concept of the chirality group of an abstract polytope, an invariant which in some sense measures the degree of mirror asymmetry (irreflexibility) of the polytope; this is an analogue of the chirality group for hypermaps introduced in [6] (see also [4]). Here we restrict ourselves to chiral or directly regular polytopes, although the concept can be generalized to more general classes of polytopes.…”

“…Here we restrict ourselves to chiral or directly regular polytopes, although the concept can be generalized to more general classes of polytopes. Our discussion is in terms of automorphism groups of polytopes, not monodromy groups as in [6], but the two approaches are equivalent at least for chiral or directly regular polytopes (in general, for other kinds of polytopes, the definitions would have to involve the monodromy group -see [23,27]). For ease of presentation we restrict ourselves to polytopes, although the concept of chirality group applies more generally to pre-polytopes (see Remark 3.1).…”

On chirality groups and regular coverings of regular oriented hypermaps

“…It follows from [4] that the covering H ∆ → H is smooth (non-branch covering), while the covering H → H ∆ may, in general, not be smooth. However, the word XY X seen as an automorphism acts like a translation ( Figure 1); so the chirality group X(H ) is δXYX δ Figure 1 generated by a translation.…”

“…See [4] for further and deeper reading on this subject. It is proved in [1] that if G = Mon(H ) has presentation x, y | R(x, y) then the chirality group of H is the normal closure of It is easy to see that both H and its ̺-dual D ̺ (H ) have the "same" monodromy group.…”