Automorphism groups of maps, hypermaps and dessins (original) (raw)

Realisation of groups as automorphism groups in permutational categories

Ars Mathematica Contemporanea, 2021

It is shown that in various categories, including many consisting of maps or hypermaps, oriented 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 them finite if A is finite. In particular, the latter applies to dessins d'enfants, regarded as finite oriented hypermaps.

Automorphism groups of maps, surfaces and

2014

Preface iii of Sciences in 2005. Many colleagues and friends of mine have given me enthusiastic support and endless helps in preparing this book. Without their helps, this book will never appears. Here I must mention some of them. On the first, I would like to give my sincerely thanks to Professor Feng Tian for his encouraging and invaluable helps and to professor Yanpei Liu introduced me into the filed of combinatorial map theory. Thanks are also given to

On the Automorphism Group of a Planar Hypermap

European Journal of Combinatorics, 1981

Several authors have investigated the problem of determining which groups can arise as automorphism groups of planar maps. The methods mostly used are topological in nature and rest upon a theorem of Kerekjarto [8] and one of Eilenberg [4] which states that a periodic homeomorphism of the sphere has exactly two fixed points. Taking any map embedded in the sphere, one can extend any automorphism of it in order to obtain a homeomorphism of the sphere. An application of the above result yields Any sense preserving automorphism of a planar map fixes exactly two cells (a cell is either a vertex or an edge or a face) .

On Automorphisms groups of Maps, Surfaces and Smarandache geometries

2011

Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automor-phisms imply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications. The first edition of this book is in fact consisting of my post-doctoral reports in Chi-nese Academy of Sciences in 2005, not self-contained and not suitable as a textbook for graduate students. Many friends of mine suggested me to extend it to a textbook for graduate students in past years. That is the initial motivation of this editi...

Automorphism Groups of Maps, Surfaces

2005

A combinatorial map is a connected topological graph cellularly embedded in a surface. This monograph concentrates on the automorphism group of a map, which is related to the automorphism groups of a Klein surface and a Smarandache manifold, also applied to the enumeration of unrooted maps on orientable and non-orientable surfaces. A number of results for the automorphism groups of maps, Klein surfaces and Smarandache manifolds and the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are discovered. An elementary classification for the closed s-manifolds is found. Open problems related the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries are also presented in this monograph for the further applications of the combinatorial maps to the classical mathematics.

Combinatorial categories and permutation groups

Ars Mathematica Contemporanea, 2015

The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group Γ, with quotient group isomorphic to Γ/N. It is shown how to enumerate such objects with a given finite automorphism group G, how to represent them all as quotients of a single regular object U(G), and how the outer automorphism group of Γ acts on them. Examples constructed include kaleidoscopic maps with trinity symmetry.

Automorphism groups of maps, surfaces and Smarandache geometries

2005

A combinatorial map is a connected topological graph cellularly embedded in a surface. This monograph concentrates on the automorphism group of a map, which is related to the automorphism group of a Klein surface and a Smarandache manifold, also applied to the enumeration of unrooted maps on orientable and non-orientable surfaces. A number of results for the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are discovered. An elementary classification for the closed s-manifolds is found. Open problems related the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries are also presented in this monograph for the further application of the combinatorial maps to the classical mathematics.

On the automorphism group of homogeneous structures

Filomat, 2020

A relational structure A with a countable universe is defined to be homogeneous iff every finite partial isomorphism of A can be extended to an automorphism of A. Endow the universe of A with the discrete topology. Then the automorphism group Aut(A) of A becomes a topological group (with the subspace topology inherited from the suitable topological power of the discrete topology on A). Recall, that a tuple 0 , ..., n−1 of elements of Aut(A) is defined to be weakly generic iff its diagonal conjugacy class (in the group theoretic sense) is dense in the topological sense, and further, the 0 , ..., n−1-orbit of each a ∈ A is finite. Investigations about weakly generic automorphisms have model theoretic origins (and reasons); however, the existence of weakly generic automorphisms is closely related to interesting results in finite combinatorics, as well. In this work we survey some connections between the existence of weakly generic automorphisms and finite combinatorics, group theory and topology. We will recall some classical results as well as some more recently obtained ones.

Polytopes with Preassigned Automorphism Groups

2015

We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.