An update on Hurwitz groups

Abstract

A Hurwitz group is any non-trivial finite quotient of the .2; 3; 7/ triangle group, that is, any non-trivial finite group generated by elements x and y satisfying x 2 D y 3 D .xy/ 7 D 1. Every such group G is the conformal automorphism group of some compact Riemann surface of genus g > 1, with the property that jGj D 84.g 1/, which is the maximum possible order for given genus g. This paper provides an update on what is known about Hurwitz groups and related matters, following up the author's brief survey in Bull. Amer. Math. Soc. 23 (1990).

Deformation theory and finite simple quotients of triangle groups I

“…This is for the positive side. On the negative side we essentially prove a conjecture of Marion proposed in [21]: in that paper he studied (a, b, c)-generation of finite quasisimple (2,4,6), (2,6,6), (2, 6, 10) (3,4,4), (3,6,6), (4,6,12) 5,7,8,9,10,11,13,15,16,17,19,22,23,25,29,31, 37, 43} (r ≥ 4) (2,3,8) r ∈ {4, 5, 7, 9, 10, 11, 13, 17, 19, 25} (2,3,9) r ∈ {4, 5, 7, 10, 11, 13, 19} (2,3,10) r ∈ {4, 5, 7, 11, 13} (2,3,11) r ∈ {4, 5, 7, 13} (2,3,12) r ∈ {4, 5, 7, 13} (2, 3, c), c ≥ 13 r ∈ {4, 5, 7} (2,4,…”

“…In a second paper, we will exploit the method further to get stronger results, paying the price of being more technical. In particular in [16] we show that the six possibly exceptional hyperbolic triangle groups in Table 1, namely those in S = {T 2,4,6 , T 2,6,6 , T 2,6,10 , T 3,4,4 , T 3,6,6 , T 4,6,12 } are not really exceptions.…”

“…A considerable effort has been made to try to classify the finite (2, 3, 7)-groups, also referred as Hurwitz groups; for a recent survey, see [3]. Recently, attention has also been given to other hyperbolic triples (a, b, c), see for example [20,21,18,22,23,24,25,9,10,12], where deterministic and probabilistic results on (a, b, c)-generation of finite simple groups of Lie type are obtained mainly in the special case where a, b and c are prime numbers.…”

Different Approaches to the Global Periodicity Problem

“…Consequently, they preserve a couple of fibrations of the plane, which are generically transversal, and given by algebraic curves. A theorem of Hurwitz of 1893, states that any compact Riemann surface with genus g > 1 admits at most 84(g − 1) conformal automorphisms, that is, homeomorphisms of the surface onto itself which preserve the local structure; see [21,22]. In our context, Hurwitz theorem can be stated as (see [33]):…”

Mirrors of reflections of regular maps

“…See [2,3,4,5,13] for further details. It follows from Theorem 5.1 that if M is a Hurwitz map of genus g with link index , then M is bounded above by 21(g − 1).…”