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.

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An orientably regular hypermap is totally chiral if it and its mirror image have no non-trivial common quotients. We classify the totally chiral hypermaps of genus up to 1001, and prove that the least genus of any totally chiral hypermap is 211, attained by twelve orientably regular hypermaps with monodromy group A 7 and type (3, 4, 4) (up to triality). The least genus of any totally chiral map is 631, attained by a chiral pair of orientably regular maps of type {11, 4}, together with their duals; their monodromy group is the Mathieu group M 11 . This is also the least genus of any totally chiral hypermap with non-simple monodromy group, in this case the perfect triple covering 3 . A 7 of A 7 . The least genus of any totally chiral map with non-simple monodromy group is 1457, attained by 48 maps with monodromy group isomorphic to the central extension 2 . Sz(8).

Bi-orientable maps (also called pseudo-orientable maps) were introduced by Wilson in the 1970s to describe non-orientable maps with the property that opposite orientations can consistently be assigned to adjacent vertices. In contrast to orientability, which is both a combinatorial and topological property, bi-orientability is only a combinatorial property. In this paper we classify the bi-orientable maps whose localorientation-preserving automorphism groups act regularly on arcs, called here bi-rotary maps, of negative prime Euler characteristic. Unlike other classification results for highly symmetric maps on such surfaces, we do not use the Gorenstein-Walter result on the structure of groups with dihedral Sylow 2-subgroups. Mathematics Subject Classification. 05C10, 05C15, 05C25.

A hypermap is (hypervertex-) bipartite if its hypervertices can be 2-coloured in such a way that "neighbouring" hypervertices have different colours. It is bipartite-uniform if within each of the sets of hypervertices of the same colour, hyperedges and hyperfaces, elements have common valencies. The flags of a bipartite hypermap are naturally 2-coloured by assigning the colour of its adjacent hypervertices. A hypermap is bipartite-regular if the automorphism group acts transitively on each set of coloured flags. If the automorphism group acts transitively on the set of all flags, the hypermap is regular. In this paper we classify the bipartite-uniform hypermaps on the sphere (up to duality). Two constructions of bipartite-uniform hypermaps are given. All bipartite-uniform spherical hypermaps are shown to be constructed in this way. As a by-product we show that every bipartiteuniform hypermap H on the sphere is bipartite-regular. We also compute their irregularity group and index, and also their closure cover H ∆ and covering core H ∆ .

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a b s t r a c tPseudo-orientable maps were introduced by Wilson in 1976 to describe non-orientable regular maps for which it is possible to assign an orientation to each vertex in such a way that adjacent vertices have opposite orientations. This property extends naturally to nonorientable and orientable hypermaps. In this paper we classify the regular pseudo-oriented maps and hypermaps of characteristic χ −3. With the help of GAP (The GAP group, 2014) and its library of small groups, we extend the classification down to characteristic χ = −16 (Tables 7-19 in the Appendix).

In this paper, we show that for any finite field [Formula: see text], any pair of map-generators (that is when one of the generators is an involution) of [Formula: see text] and [Formula: see text] has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group [Formula: see text] or [Formula: see text] is reflexible, or equivalently, there are no chiral regular maps with automorphism group [Formula: see text] or [Formula: see text]. As remarked by Leemans and Liebeck, also [Formula: see text] and [Formula: see text] are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.