Maximal automorphism groups of symmetric Riemann surfaces with small genus

AbstractThe genus of a finite group G is the smallest genus of its Cayley graphs. If G has genus g > 1, then by theorems of Tom Tucker ¦G¦⩽ 168(g − 1) and this inequality is strict unless G can be generated by elements a, b, c satisfying a2 = b2 = c2 = (ab)2 = (bc)3 = (ac)7 = 1 with ab and bc generating a proper subgroup of G. Conversely, any group G of the latter sort has genus g = ¦G¦/168 + 1, and, moreover, is faithfully representable as a group of homeomorphisms of a compact Riemann surface S of the same genus g, with half the elements of G reversing the surface's orientation. This paper describes all such groups G which have order less than 2 million, and gives for each corresponding value of g (in the range 1 < g < 11,905) the number of distinct possibilities for the Riemann surface S. The classification follows that of all Hurwitz groups of order less than 1 million, obtained by the same author