On Hurwitz groups of low rank
نویسندگان
چکیده
Let , 3, 7) denote the infinite triangle group, defined by the presentation X,Y | X = Y 3 = (XY ) = 1 . A non-trivial group G is said to be (2, 3, 7) -generated (or a Hurwitz group, when finite) if it is an epimorphic image of , 3, 7). Hurwitz groups are particularly interesting for the theory of Riemann surfaces. Namely, if H is the automorphism group of a compact Riemann surface of genus g > 1, then H is finite of order not exceeding 84(g− 1), and the Hurwitz groups are exactly those for which |H| = 84(g− 1). As a quotient of a Hurwitz group is again Hurwitz, it is crucial to determine the simple Hurwitz groups. The values of q for which the groups PSL(2, q) are Hurwitz were determined by A. M. Macbeath in 1969 ([Mac]), whereas J. Cohen in 1981 ([Coh]) showed that PSL(3, 2) is the only Hurwitz group in the series PSL(3, q), and none of the groups in the series PSU(3, q) are Hurwitz. M. Conder in 1980 ([Con]) proved that the alternating groups An are Hurwitz, provided n > 167. More recently G. Malle has shown that the exceptional simple groups G2(q), q 5; G2(q), q > 3; D4(q) for q = p , p = 3, q = 4, and F4(2 ) for m 1 (3) are Hurwitz ( cfr. [Mal1], [Mal2]). The sporadic Hurwitz groups are also known, with the exception of the Monster. (For a bibliography and related comments, see [Jon]). Recent constructive results of Lucchini, Tamburini and Wilson ([LTW], [Lu], [Wil] ) show that the family of (2, 3, 7) -generated groups, and more generally of (2, 3, k) -generated groups, is very large.
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