On symmetric representations of groups of automorphism of bordered Klein surfaces
A Klein surface is a compact topological surface with a dianalytic structure. A bordered Klein surface \(X\) of algebraic genus \(g\geq2\) can be represented as a quotient \(X=\mathcal{H}/\Gamma\), where \(\mathcal{H}\) denotes the hyperbolic plane and \(\Gamma\) is an non-euclidean crystallographic group (NEC group, in short). The group \(\Gamma\) has a signature \((g',\pm,[-],\{(-)^k \})\), where \(k>0\) is the number of boundary components of \(X\), the algebraic genus of \(X\) is \(g=\epsilon g'+k-1\), where \(\epsilon=2\) if \(X\) is an orientable surface (sign `\(+\)' in the above signature) or \(\epsilon =1\) if \(X\) is a non-orientable surface (sign `\(-\)'). A finite group \(G\) is a group of automorphisms of \(X=\mathcal{H}/\Gamma\) if there exists an NEC group \(\Lambda\) and an epimorphism \(\theta: \Lambda \rightarrow G\) such that \(\Gamma= \ker\theta\) and \(G=\Lambda/\Gamma\). An automorphism of \(X\) induces a permutation of its boundary components, \(\mathcal{O}_1, \dots, \mathcal{O}_k\). In the paper under review, the authors study the corresponding groups of representations of groups of automorphisms of \(X\) in the corresponding finite symmetric groups \(S_k\). Let \(\rho_X:G\rightarrow S_k\) be given by \(\rho_X(g)(i)=j \Leftrightarrow g(\mathcal{O}_i)=\mathcal{O}_j\). This representation is called geometric. The authors prove that \(\rho_X\) is a homomorphism whose kernel is either cyclic or dihedral. Next, they study this kernel. Given a set of canonical generators of \(\Lambda\), let \(\mathcal{C}\) be the set of its canonical reflections. Let \(\{c_i : i\in I\}\) be a maximal subset of \(\mathcal{C}\) consisting of the pairwise non-conjugate reflections, mapped by \(\theta\) into the unit. Then it is proved that \[ \ker\rho_X=\bigcap_{g\in G} \bigcap_{i\in I}g\big(\theta(C(\Lambda,c_i))\big)g^{-1} \] where \(C\) denotes the centralizer group. In the main result, the authors prove that an abstract representation \(\rho:G\rightarrow S_k\) is equivalent to the representation of a group of automorphisms of some Klein surface on its set of boundary components if and only if \(\rho\) is the product of regular representations with respect to cyclic or dihedral subgroups. Given a finite group \(G\) there exists a bordered Klein surface \(X\) such that \(G\) is a group of automorphisms of \(X\) and \(\rho_X\) is faithful. The authors consider the problem of finding the minimal \(k\), called minimal degree \(k(G)\), and they obtain a formula. Since \(k(G)\leq |G|\), the authors look for groups for which \(k(G)=|G|\). They study several examples and prove that the equality holds if and only if \(G\) is one of the following groups: (a) A cyclic group of order a power of a prime. (b) A generalized quaternion group. (c) An elementary abelian group of order \(2^n\).
scientific article; zbMATH DE number 6132872