Lifting map automorphisms and MacBeath's theorem (Q1813618)

A map \(M\) consists of a 2-cell imbedding of a connected graph \(\Gamma\) into a closed orientable 2-manifold \(S_ g\) of genus \(g\). A map automorphism is a graph automorphism preserving oriented region boundaries; \(\Aut M\) is the associated group. Always \(|\Aut M|\leq 2| E(\Gamma)|\); if equality holds \(M\) is said to be symmetrical. For \(g>1\), \(|\Aut M|\leq 84(g-1);\) if equality holds \(M\) is said to be extremal. A connection is given by: For \(M\) symmetrical and \(g>1\), \(M\) is extremal if and only if \((d,k)=(7,3)\) or \((3,7)\), where \(d\) is the common vertex degree and \(k\) is the common face degree [\textit{N. L. Biggs} and \textit{A. T. White}, Permutation groups and combinatorial structures (1979; Zbl 0415.05002), p. 132]. Also relevant is the theorem of \textit{A. M. Macbeath} [Proc. Sympos. Pure Math. XII, 1969], that there are infinitely many finite extremal maps. In the present paper the author gives a cohomological covering construction for maps which preserves both the symmetrical and the extremal properties. Starting with the ''Klein map'' \(M\) \((\Aut M=PSL(2,7)\), \(g=3\), \((d,k)=(3,7)\) -- or \((7,3)\) for the dual), which is both symmetrical and extremal, and iterating the covering construction, proves Macbeath's theorem anew.