Finite groups acting on bordered surfaces and the real genus of a group
The author introduces the notion of real genus \(\rho(G)\) of a finite group \(G\) as the minimum algebraic genus of any compact bordered surface on which \(G\) acts as a group of homeomorphisms. As a consequence of the Riemann-Hurwitz formula, the author proves that if \(o(G)\) is the order of a finite group generated by \(r\) elements of orders \(m_ 1,\dots,m_ r\), then \[ \rho(G)\leq 1+ o(G) \left[ r-1+ \sum^ r_{i=1} 1/m_ i\right]. \] As the main result, the author computes the real genus of several important groups as \(\text{PSL}(2,q)\) for many values of \(q\), \(S_ n\), \(A_ n\) for \(n\geq 168\), \((\mathbb{Z}_ 2)^ n\), \((\mathbb{Z}_ 3)^ n\) and the dicyclic groups \(G_ n\). He also rewrite in a very elegant way, in terms of the real genus, some already known results. For example he determines the groups with prescribed real genus \(\leq 3\), and by using [\textit{E. Bujalance} and \textit{G. Gromadzki}, Proc. Am. Math. Soc. 108, 749-759 (1990; Zbl 0696.30047)] and [the author, Proc. Am. Math. Soc. 101, 287-292 (1987; Zbl 0626.30044)], the following inequalities are proved: \[ \begin{aligned} o(G) &\leq 8(\rho(G)- 1)\text{ if } G\text{ is nilpotent with }\rho(G)\geq 2,\\ o(G) &\leq {p\over p-2} (\rho(G)- 1)\text{ if }G\text{ is a \(p\)-group},\;p\neq 2,\;\rho(G)\geq 2.\end{aligned} \] The notion of symmetric genus \(\sigma(G)\) of a finite group \(G\) is defined as the real genus but dealing with compact, orientable unbordered surfaces. If \(G\) acts on a bordered surface whose boundary has \(k\) connected components, the author proves using two classical topological constructions the inequalities \[ \begin{aligned} 2\sigma(G) &\leq \rho(G)- k+ 1\text{ if \(X\) is orientable}\\ \sigma(G) &\leq \rho(G)- k\text{ if \(X\) is non-orientable}\end{aligned} \] and so \(\sigma(G)< \rho(G)\) if \(\rho(G)\neq 0\).
scientific article; zbMATH DE number 515199
scientific article