Page 333 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 333
SYMMETRY OF GRAPHS, MAPS AND POLYTOPES (MS-9)
Non-Cayley regular maps and generalizations of skew-morphisms
Robert Jajcay, robert.jajcay@fmph.uniba.sk
Faculty of Mathematics, Physics and Informatics of the Comenius University,
Bratislava, Slovakia
A Cayley map CM (G, X, p) is a 2-cell embedding of a connected Cayley graph C(G, X) into
an orientable surface having the property that for each a ∈ G the graph automorphism of
C(G, X) induced by the left multiplication by a, g → a · g, ‘lifts’ into a map automorphism
of the embedding. Hence, all Cayley maps admit automorphism groups acting regularly on the
vertices of the map, namely, the groups GL ∼= G of automorphisms induced by left multipli-
cations by the elements of G. As is well-known, an embedding of a Cayley graph C(G, X)
into an orientable surface is a Cayley map if and only the rotation system of the embedding
has the property that all local permutations of the neighbors gx, x ∈ X, of the vertices g ∈ G
determined by the embedding are equal to a constant cyclic permutation p of X.
A 2-cell embedding of a connected graph into an (orientable or non-orientable) surface,
a map M, is said to be regular if the full automorphism group of M acts regularly on its
flags. An orientable map is said to be orientably regular if it admits an orientation preserving
automorphism group acting transitively on its arcs. A Cayley map CM (G, X, p) is known to
be orientably regular if and only if there exists a skew-morphism of G that agrees on X with p.
In our talk, we will focus on orientably regular and regular maps which are not Cayley, but are
in some sense ‘close to being Cayley’ and which admit a ‘partial’ skew-morphism generating a
vertex-stabilizer of the map. The ideas stem from a recent article on generalized Cayley maps
written jointly by the presenter, J. Siran and Y. Wang, and from the concept of half-regular
Cayley maps introduced by the presenter and R. Nedela.
Realisation of groups as automorphism groups of maps and hypermaps
Gareth A. Jones, G.A.Jones@maths.soton.ac.uk
University of Southampton, United Kingdom
I will show that in various categories, including many consisting of maps or hypermaps, ori-
ented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological space,
every countable group A is isomorphic to the automorphism group of uncountably many non-
isomorphic objects, infinitely many of which are finite if A is finite. In particular, the latter
applies to dessins d’enfants, regarded as finite oriented hypermaps. The objects realising A are
obtained as regular coverings by A of certain basic objects with primitive monodromy groups,
corresponding to maximal subgroups of triangle groups. The constructions of these generalise
results of Bernhard Neumann on maximal subgroups of infinite index in the modular group, and
of Marston Conder on maximal subgroups of finite index in various cocompact triangle groups.
331
Non-Cayley regular maps and generalizations of skew-morphisms
Robert Jajcay, robert.jajcay@fmph.uniba.sk
Faculty of Mathematics, Physics and Informatics of the Comenius University,
Bratislava, Slovakia
A Cayley map CM (G, X, p) is a 2-cell embedding of a connected Cayley graph C(G, X) into
an orientable surface having the property that for each a ∈ G the graph automorphism of
C(G, X) induced by the left multiplication by a, g → a · g, ‘lifts’ into a map automorphism
of the embedding. Hence, all Cayley maps admit automorphism groups acting regularly on the
vertices of the map, namely, the groups GL ∼= G of automorphisms induced by left multipli-
cations by the elements of G. As is well-known, an embedding of a Cayley graph C(G, X)
into an orientable surface is a Cayley map if and only the rotation system of the embedding
has the property that all local permutations of the neighbors gx, x ∈ X, of the vertices g ∈ G
determined by the embedding are equal to a constant cyclic permutation p of X.
A 2-cell embedding of a connected graph into an (orientable or non-orientable) surface,
a map M, is said to be regular if the full automorphism group of M acts regularly on its
flags. An orientable map is said to be orientably regular if it admits an orientation preserving
automorphism group acting transitively on its arcs. A Cayley map CM (G, X, p) is known to
be orientably regular if and only if there exists a skew-morphism of G that agrees on X with p.
In our talk, we will focus on orientably regular and regular maps which are not Cayley, but are
in some sense ‘close to being Cayley’ and which admit a ‘partial’ skew-morphism generating a
vertex-stabilizer of the map. The ideas stem from a recent article on generalized Cayley maps
written jointly by the presenter, J. Siran and Y. Wang, and from the concept of half-regular
Cayley maps introduced by the presenter and R. Nedela.
Realisation of groups as automorphism groups of maps and hypermaps
Gareth A. Jones, G.A.Jones@maths.soton.ac.uk
University of Southampton, United Kingdom
I will show that in various categories, including many consisting of maps or hypermaps, ori-
ented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological space,
every countable group A is isomorphic to the automorphism group of uncountably many non-
isomorphic objects, infinitely many of which are finite if A is finite. In particular, the latter
applies to dessins d’enfants, regarded as finite oriented hypermaps. The objects realising A are
obtained as regular coverings by A of certain basic objects with primitive monodromy groups,
corresponding to maximal subgroups of triangle groups. The constructions of these generalise
results of Bernhard Neumann on maximal subgroups of infinite index in the modular group, and
of Marston Conder on maximal subgroups of finite index in various cocompact triangle groups.
331
