Highly symmetric maps and dessins (Q2810318)

In this nice (but compact) book, the subject matter is maps on surfaces and areas together with their automorphism groups. The author concentrates on the most symmetric objects in these categories and gives techniques for enumerating and classifying them in terms of their embedded graph, their underlying surface and their automorphism group. Applications to other areas such as abstract polytopes and covering space theory are given.NEWLINENEWLINEThe book consists of 9 chapters, 4 appendices covering necessary background and a good bibliography.NEWLINENEWLINEIn Chapter 1 Oriented maps and permutations are considered. A map \(\mathcal{M}\) is an embedding of a graph \(\mathcal{G}\) in a surface \(S\) without crossings, so that the faces (connected components of \(S \setminus \mathcal{G}\)) are simply connected, i.e., homeomorphic to an open disc. It is assumed that the surface \(S\) is orientable with a chosen orientation. Given an oriented map \(\mathcal{M}\), let \(\Phi\) be the set of its arcs (directed edges). Define \(x\) to be the permutation of \(\Phi\) which follows the chosen orientation around each incident vertex so that the cycles of \(x\) correspond to the vertices of \(\mathcal{M}\) with the length of each cycle equal to the valency of the corresponding vertex. Define \(y\) to be the permutation which reverses the direction of each arc so that the cycles of \(y\) correspond to the edges of \(\mathcal{M}\).NEWLINENEWLINEThe permutation group \(G = \langle x, y\rangle \leq \mathrm{Sym }\Phi\) generated by \(x\) and \(y\) is called the monodromy group of \(\mathcal{M}\). The (orientation preserving) automorphisms of \(\mathcal{M}\) are the permutations of \(\Phi\) which commute with \(x\) and \(y\). So, the automorphism group \(A = \mathrm{Aut }M\) is identified with the centralizer \(C(G)\) of \(G\) in \(\mathrm{Sym }\phi\). It is assumed that the graph \(\mathcal{G}\) and surface \(S\) are connected. Equivalently, it is assumed that \(G\) is transitive on \(\Phi\). Thus, one can study all oriented maps by considering the orientably regular maps and their automorphism groups.NEWLINENEWLINEThe following theorem is proved.NEWLINENEWLINETheorem. Every oriented map \(\mathcal{M}\) is isomorphic to the quotient \(N / S\) of an orientably regular map \(N\), of the same type, by a subgroup \(S \leq\mathrm{Aut }N\). If \(\mathcal{M}\) is compact, then \(N\) can also be chosen to be compact.NEWLINENEWLINEIn Chapter 2, the classification of orientably regular maps in terms of their automorphism groups are studied. It is shown that the complete graph \(K_n\) has an orientably regular embeddings if and only if \(n\) is a prime power. If \(n = p^{e}\), where \(p\) is a prime, then there are \(\phi(n - 1) / e\) orientably regular embeddings of \(K_n\).NEWLINENEWLINEIn Chapter 3, the classification of orientably regular maps by their surface or equivalently, in the compact case, by their genus is considered.NEWLINENEWLINEThe following theorems are proved.NEWLINENEWLINETheorem. If \(A\) is the automorphism group of an orientably regular map \(\mathcal{M}\) of genus \(g > 1\), then \(|A| \leq 84 (g - 1)\), attained if and only if \(\mathcal{M}\) has type \(\{3, 7\}\) or \(\{7, 3\}\).NEWLINENEWLINETheorem. For each integer \(g \geq 2\) there is an orientably regular map of genus \(g\) with an orientation-preserving automorphism group of order \(g(g + 1)\).NEWLINENEWLINEIn Chapters 4 and 5, permutational categories are studies and their examples are given. A hypermap is an embedding of a hypergraph \(\mathcal{G}\) in a surface. Exact values of genus of certain orientably regular hypermaps with orientation preserving automorphism group are given.NEWLINENEWLINEIn Chapter 6, counting of regular objects are considered. The main theorem is as follows.NEWLINENEWLINETheorem. If \(\mathfrak{G}\) is a permutational category with a finitely generated parent group \(\Gamma\) and \(G\) is a finite group, then the number \(r(G)\) of isomorphism classes of regular objects \(O \in \mathfrak{G}\) with \(\mathrm{Aut }O \cong G\) is given by \(r(G) = \dfrac{1}{|\mathrm{Aut}(G)|} \sum_{H \leq G}\mu_{G}(H)|\mathrm{Hom }(\Gamma, H)|\). The number \(m(G)\) of isomorphism classes of objects \(O \in \mathfrak{G}\) with \(\mathrm{Mon }O \cong G\) is given by \(m(G) = r(G) c(G)\), where \(c(G)\) is the number of conjugacy classes of subgroups of \(G\) with trivial core.NEWLINENEWLINEIn Chapter 7, evaluation of Möbius function is studied. The Möbius function is calculated for some infinite families.NEWLINENEWLINEIn Chapter 8, counting of homomorphisms is considered. The following theorems are proved.NEWLINENEWLINETheorem. Let \(X\), \(Y\) and \(Z\) be conjugacy classes in a finite group \(H\). Then, the number of solutions of the equation \(xyz = 1\) in \(H\), with \(x \in X\), \(y \in Y\) and \(z \in Z\) is \(\dfrac{|X||Y||Z|}{|H|}\sum\dfrac{\chi(x)\chi(y)\chi(z)}{\chi(1)}\) where \(x \in X\), \(y \in Y\) and \(z \in Z\) and \(\chi\) ranges over the irreducible complex characters of \(H\).NEWLINENEWLINETheorem. If \(H\) is any finite group, then \(|\mathrm{Hom}(\prod_{g}, H)| = |H|^{2g - 1}\sum\chi(1)^{2 - 2g}\), where \(\chi\) ranges over the irreducible complex characters of \(H\).NEWLINENEWLINEIn Chapter 9, it is shown that maps can provide link between other areas of mathematics such as Riemann surface theory, algebraic geometry, algebraic number theory and Galois theory.