Edge transitive maps

A {\em map} is a $2$-cell embedding of a graph into a closed surface, denoted by $\calM=(V, E, F)$ a map with vertex set $V$, edge set $E$, and face set $F$. An {\em automorphism} of $\calM$ is a permutation on $V\cup E\cup F$ which preserves $V$, $E$ and $F$, and their incidences. Under composition of permutations, all automorphisms form the {\em automorphism group} $\Aut(\calM)$. An {\em edge-transitive} map $\calM$ has a high symmetry degree; namely, the automorphism group $\Aut(\calM)$ is transitive on the edge set. Studying edge-transitive maps has been a central topic in algebraic and topological graph theory for a long time. In particular, Graver and Watkins classified automorphism groups of edge-transitive maps into fourteen types (\emph{Mem. Amer. Math. Soc.}, 1997). Automorphism groups of maps in each type correspond to a unique algebraically consistent combination of stabilizers of vertices, edges, faces, and Petrie walks. Among the fourteen types, one type is (flag) {\em regular}, six types are {\em arc-regular} up to duality, and the others are {\em edge-regular}. The {\em Euler characteristic} is a topological invariant for a map $\calM$, which is equal to $|V|-|E|+|F|$. Breda, Nedela, and {\v{S}}ir{\`{a}}{\v{n}} classified regular maps of negative prime Euler characteristic (\emph{Trans. Amer. Math. Soc.}, 2005). It is shown that the Euler characteristic of a map affects the automorphism group. As a generalization, we initiate the study of the following problem in this thesis: \emph{Classify edge-transitive maps such that the Euler characteristic is prime to the edge number}. The main results of this thesis include the following three parts. \begin{enumerate}[\rm(i)] \item~ It is proved that there exist exactly thirteen types of edge-transitive maps whose Euler characteristic is prime to the edge number. Examples of each of the thirteen types are constructed. The unique type in the exception is one of the seven edge-regular types. \item~ A classification is given of arc-transitive maps, of which automorphism groups are non-solvable, and the Euler characteristic is prime to the edge number. In this classification, maps are face-intransitive and arc-regular, except for regular maps. In addition, the stabilizers of the points and faces in the automorphism group of the map in this classification result are maximal dihedral subgroups. The enumeration of isomorphic classes of face-intransitive and arc-regular maps is provided. \item~A characterization of orientable maps whose Euler characteristic is prime to the edge number is given. Automorphism groups of such maps are metacyclic or contain a cyclic subgroup of index at most $4$. The later case only occurs when it comes to automorphism groups of regular maps. There exist exactly eight types of orientable edge-transitive maps whose Euler characteristic is prime to the edge number.

    地图是一个图在曲面上的双胞腔嵌入。    我们通常将一个地图$\calM$记作$(V,E,F)$, 其中$V$、$E$、$F$ 分别是地图的点集、边集和面集。
    一个地图的自同构是集合$V\cup E\cup F$上的一个保持点集、边集和面集不变，且保持点、边和面间的邻接关系不变的置换。   所有地图的自同构在置换的复合关系下构成地图的自同构群。   如果一个地图的自同构群在其边集上传递，那么我们称其为一个边传递地图。   边传递地图是一类具有很高对称性的地图。    
    长久以来，对边传递地图的研究一直是代数图论和拓扑图论中的一个热点。    其中一个重要的成果是Graver和Watkins给出的对边传递地图及其自同构群的十四类划分（\emph{Mem. Amer. Math. Soc.}, 1997）。
    每一类中，地图的自同构群对应于唯一的一个代数上相容的点、边、面和Petrie路的稳定子的组合。    在这十四类中，一类是（旗）正则地图，六类在对偶意义下是弧正则地图，其余的七类是边正则地图。    
    一个地图的欧拉示性数是它的一个拓扑不变量，其被定义为点数加面数减去边数，与其曲面的欧拉示性数一致。    Breda，Nedela，和{\v{S}}ir{\`{a}}{\v{n}}分类了欧拉示性数为负素数的正则地图（\emph{Trans. Amer. Math. Soc.}, 2005）。    这篇文章指出，地图的欧拉示性数会对其自同构群做出限制。
    作为以上结果的推广，这篇论文率先开始了对如下问题的研究：分类欧拉示性数和边数互素的边传递地图。
        
    这篇论文的主要结果分为以下三个部分。\begin{enumerate}[\rm(i)]
    \item~在十四类边传递地图中，我们证明了存在十三类欧拉示性数和边数互素的边传递地图。    十四类边传递地图中唯一的例外属于一类边正则地图。
    我们对十三类中的每一类地图构造了例子。    \item~完全分类了自同构群为不可解群的，且欧拉示性数和边数互素的弧传递地图。    其中，除了正则地图以外，地图是弧正则且面不传递的。    %其中这唯一一个正则地图是Petersen图在射影平面上的嵌入。
    另外，在这个分类结果中，地图的点、面在自同构群中的稳定子都是极大二面体子群。    对于弧正则且面不传递的地图，我们计数了其同构类的个数。    \item~刻画了欧拉示性数和边数互素的可定向地图。    这类地图的自同构群是亚循环群或者包含一个指数至多为四的循环子群。    后一种情况只发生在正则地图的自同构群中。    存在八类可定向的，欧拉示性数和边数互素的边传递地图。

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