Hurwitz Groups of Large Rank

Abstract

A finite non-trivial group G is called a Hurwitz group if it is an image of the infinite triangle group Δ:=Δ(2,3,7)=〈X, Y|X2=Y3=(XY)7=1〉 Thus G is a Hurwitz group if and only if it can be generated by an involution and an element of order 3 whose product has order 7. The history of Hurwitz groups dates back to 1879, when Klein [9] was studying the quartic x3y+y3z+z3x=0 of genus 3. The automorphism group of this curve has order 168 = 84(3−1), and it is isomorphic to the simple group PSL2(7), which is generated by the projective images of the matrices x : = ( 0 1 - 1 0 ) , y : = ( 0 - 1 1 - 1 ) with product x y = ( 1 - 1 0 1 ) and so is a Hurwitz group. In 1893, Hurwitz [7] proved that the automorphism group of an algebraic curve of genus g (or, equivalently, of a compact Riemann surface of genus g) always has order at most 84(g−1), and that, moreover, a finite group of order 84(g−1) can act faithfully on a curve of genus g if and only if it is an image of Δ(2, 3, 7). The problem of determining which finite simple groups are Hurwitz groups has received considerable attention. In [10], Macbeath classified the Hurwitz groups of type PSL2(q); there are infinitely many of them. In [1] Cohen proved that no group PSL3(q) is a Hurwitz group except PSL3(2), which is isomorphic to PSL2(7). Certain exceptional groups of Lie type, and some of the sporadic groups, are known to be Hurwitz groups. For discussions of the results on these groups we refer the reader to [3, 5, 11].