Symmetrical maps arising from regular Coxeter elements of linear groups (Q1111565)

Let \(\Gamma\) be the group \(SL(2n+1,q)\), where q is an odd prime power, with trivial center. Taking \(\Lambda\) and \(\Lambda\) ' to be conjugacy classes of \(\Gamma\), the author constructs a finite connected graph \(G=G(V,E)\) with vertex set \(V=\Lambda\) and edges \((\gamma_ 1,\gamma_ 2)\in E\) precisely when \(\gamma_ 1\gamma_ 2\in \Lambda '.\) He proves that \(\Gamma\) acts as a group of automorphisms such that (1) each vertex stabilizer \(\Gamma_ v\), \(v\in V\), is cyclic of order \(>1\), and (2) for each \(v\in V\), \(\Gamma_ v\) acts transitively on the set of neighbors of v. There exists a map structure on G, an embedding of the underlying topological space of G into the compact orientable surface S [see \textit{N. L. Biggs} and \textit{A. T. White}, Permutation groups and combinatorial structures, Lond. Math. Soc. Lecture Notes Ser. 33 (1979; Zbl 0415.05002)]. The author is concerned with the question which integers g can serve as genus of these associated surfaces S. He reaches to the assertion that \[ 2-2g=q^{(2n+1/2)}\cdot \prod^{2n+1}_{i=2}(q^ i-1)\cdot ((q- 1)/(q^{2n+1}-1)-1/6), \] conjecturing that a class \(\Lambda\) and a symmetrical map structure on the resulting graph G can be found such that the corresponding surface S is triangulated. Some calculations supporting this conjecture are given here.