Bi--rotary Map Of Negative Prime Square Characteristic

A map is an embedding of a graph Γ into a surface S such that each connected compo-nent of S\(V ∪E)is homeomorphic to an open disk.The mathematical theory to research the maps is called map theory or topological graph theory.By viewing the vertices,edges and faces as 0 cells,1 cells and 2 cells on the carrier surface S respectively,each map has a Euler characteristic,that is,the number of vertexes minus the number of edges,and then plus the number of faces.We can also reformulate it by topology language.That is the number of 0 cells minus the number of 1 cells,and then plus the number of 2 cells.The Euler characteristic(or called Euler-Poincaré formula)is a famous theorem in classical map theory,which associates with such important branches of mathematics as polyhedra,topology and graphs.The Euler characteristic of a map is a very important character of a map because it can reveal some interesting property of the map.The Euler characteristic is a classical subject in topological graph theory.In this decades,the contents of map theory have been extended greatly.The classical viewpoint about how to deal with a map have been developed to three modern viewpoints,corresponding to topology theory,Riemann geometry theory and group theory.In the view of group theory,a map with a high level of symmetry can be represented by a group and its cosets,hence maps with a high level of symmetry are deeply associated with group theory,similar to the relationship between theory of highly symmetric graphs and group theory.An automorphism of M is a permutation on the set of flags that preserving the incidence relations.All these permutations form the automorphism group denoted by Aut(M)under the permutation composition.Aut(M)acts semi-regularly on the flag set.If the action is regular,then we call M is a regular map.Thus,Regular maps exhibit in some sense the highest level of symmetry and bi-rotary maps also exhibiting a high level of symmetry.The problem investigated in this thesis is what will happen if we loose the restriction of the symmetry slightly.There are many kinds of symmetry in map theory,such as flag regular map,arc regular map and edge regular map.Flag regular map is exactly the regular map.Such maps exhibit the highest level of symmetry,because a regular map must be an arc regular map,and hence edge regular map.An edge regular map is a map whose automorphism group acts on the edge set transitively.And an arc regular map is a map whose automorphism group acting on the arc sets transitively.This thesis studies a special kind of arc regular maps,called bi-rotary map.To define it,the concept of local orientation which is associated with topology theory is necessary and important.A map is called orientable if its carrier surface is orientable.Then we can induce a local orientation around each vertex such that when moving along any edges within a sufficiently narrow band on the carrier surface,the local orientations at the two vertices incident to the edge are consistent.Now,if a map admits an assignment of local orientations at each vertex of the map(the carrier surface is not necessarily orientable)such that when moving along the edges,the local orientations at the vertices incident to the same edge are always opposite,then this map is called bi-orientable.For a bi-orientable map,there is a special subgroup of the automorphism group Aut(M)denoted by Aut~b(M)which consists of automorphisms preserving the local orientation of M.A bi-rotary map is just a bi-orientable map with Aut~b(M)acting regularly on the arc set of M.Our work is to classify all bi-rotary map of negative prime square characteristic which connected two subjects of the map theory.The method of translating a map problem to a group problem has been proved very useful in studying highly symmetric maps.For regular maps,people made breakthrough a decade ago in a problem similar to ours.To be more specific,the classification of regular maps of negative prime Euler characteristic was obtained.This research was completed by Breda d’Azevedo,Antonio and Nedela,Roman and ?iráň,Jozef in 2005.The authors used the group method to represent a regular map and he applied some deep results in group theory.Following their idea,people have derived the classification of regular maps of negative prime square Euler characteristic and the classification of regular maps of three times of prime square Euler characteristic.In addition,thanks to the computer aids,we have derived a complete classification of regular maps of a large negative Euler charac-teristic.However,the research on bi-rotary maps has just started.In 2019,Breda and Catalano,and ?iráň published a paper which solved the classfication problem of bi-rotary maps of negative prime characteristic.In that paper,the authors avoided using those very deep results in group theory.They used the case-by-case discussion on the order of the automorphism groups and the Fitting subgroup method.This thesis follows the idea of the paper in 2019 which solved the classification prob-lem of bi-rotary map of negative prime characteristic and develops tools used in the orig-inal paper.We generalize a special case of bi-rotary maps.To be specific,we solve the classification of a special kind of non-orientable bi-rotary map whose number of the arcsis equal to the least common multiplier of the valency and the length of the close walk of the faces.This case sources from the original paper in 2019 directly and this paper uses the Fitting subgroup method.However,the paper only concentrated on the classification of the case that characteristic is a negative prime and it does not divide the order of the auto-morphism group(which is equal to the number of the arcs).We notice that their results can be generalized by Poincaré’s lemma so that we can classify all of non-orientable bi-rotary maps whose number of the arcs is equal to the least common multiplier of the valency and the length of the close walk of the faces.The key point is that the orientation-preserving automorphism group is the complex product of its vertex stabilizer and face stabilizer by Poincaré ’s lemma.Hence the whole group is a solvable group as its vertex stabilizer is cyclic and face stabilizer is dihedral.We can therefore apply the Fitting subgroup method again.By a similar inference,we derive the classification of the non-orientable maps whose number of the arcs is equal to the least common multiplier of the valency and the length of the close walk of the faces.It can be noted that this classification is independent on the Euler characteristic of the bi-rotary maps,which means we also solve a special case of non-orientable maps.The paper in 2019 indicates that whether the Euler characteristic divides the order of the bi-orientation-preserving automorphism group is an important thing.Both the re-sults and the methods used are quite different in those two different cases.This thesis also use the same case discussion method.By an elementary but non-trivial lemma,we develop the case discussion method used in the paper in 2019.The lemma indicates that we should discuss in three cases for the bi-rotary maps of negative prime square Euler characteristic.They are the case that the prime divisor of the Euler characteristic does not divide the order of the bi-orientation-preserving automorphism group,the case that the prime divisor of the Euler characteristic divides the order of bi-orientation-preserving automorphism group exactly and the case that the Euler characteristic divides the order of bi-orientation-preserving automorphism group.The lemma mentioned above is also a simple method to simplify some argument of similar problems.It shows how powerful the group method is when the Euler characteristic only have one or two prime divisors.On the other hand,it also indicates that this method may be restricted when the Euler character-istic has many prime divisors.It can be very complex.For the case that the prime divisor of the Euler characteristic does not divide the order of the bi-orientation-preserving auto-morphism group,by the lemma mentioned before,we derive that in this case the numberof the arcs of the maps is exactly the least common multiplier of the valency and the length of the close walk along the faces.Hence we can apply the classification of that special case and exclude this case.As for the case that the prime divisor of the Euler character-istic divides the number of arcs,we can derive two cases by previous lemma.One case is also that the number of the arcs is equal to the least common multiplier of the valency and the length of the close walk along the faces.It can also be excluded by applying the classification of that special case.Another case is that the number of arcs is the p times of the least common multiplier of the valency and the length of the close walk along the faces,where p is the prime divisor of the Euler characteristic of the map.In this case,we apply Gorenstein’s deep results of group theory,which describes the classification of the group with dihedral Sylow 2-subgroups.Notice that in this case,the Sylow 2-subgroup of bi-orientation-preserving automorphism group is dihedral,and the other odd Sylow sub-groups are cyclic.Hence we can apply the Gorenstein theorem naturally.Now we discuss it on the solubility of the group.If the bi-orientation-preserving automorphism group is solvable,we can induce a contradiction by the Fitting group method.If the group is un-solvable,then we can also induce a contradiction by Gorenstein theorem.Hence all cases are impossible.Only one case left.That is the case the Euler characteristic divides the or-der of the bi-orientation-preserving automorphism group.This case is similar to the case in the paper in 2019.It has a finite possibility,which means only finite choices of the valency and the length of the close walk alone the faces are reasonable.By the analysis similar to the paper in 2019,we derive that there are only two bi-rotary maps can have an odd negative prime square characteristic and the prime is 3.As for the even prime p = 2,the classification has been contained in Breda d’Azevedo,Antonio and Catalano,Domenico A.and Duarte,Rui’s paper in 2015.Hence there are finite kinds of the bi-rotary maps of the negative prime square characteristic,which is different from the classification of the bi-rotary maps of the negative prime characteristic.