Simple group actions on hyperbolic Riemann surfaces of least area

S. Allen Broughton

Vol. 158 (1993), No. 1, 23–48

DOI: 10.2140/pjm.1993.158.23

Abstract

It is shown that if a simple group G acts conformally on a hyperbolic surface of least area (or alternatively, on a Riemann surface of least genus σ ≥ 2), then G is normal in Aut(S) and the map Aut(S) → Aut(G) induced by conjugation is injective. For the preponderance of these minimal actions the group Aut(S)∕G is isomorphic to a subgroup of Σ₃. It is shown how to compute Aut(S) purely in terms of the group-theoretic structure of G, in these cases. As examples and as part of the proof, the minimal actions and the groups Aut(S) are completely worked out for A₅, SL₃(3), M₁₁ and M₁₂.

Mathematical Subject Classification 2000

Primary: 57M60

Secondary: 20F38, 30F10

Milestones

Received: 1 February 1990

Revised: 20 December 1991

Published: 1 March 1993