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Diffeomorphisms and automorphisms of compact hyperbolic 2-orbifolds. (English) Zbl 1198.30039

Gardiner, Frederick P. (ed.) et al., Geometry of Riemann surfaces. Proceedings of the Anogia conference to celebrate the 65th birthday of William J. Harvey, Anogia, Crete, Greece, June–July 2007. Cambridge: Cambridge University Press (ISBN 978-0-521-73307-6/pbk). London Mathematical Society Lecture Note Series 368, 139-155 (2010).
The main result proved in this paper is that if \(X\) is a compact hyperbolic two-orbifold, then \(\mathrm{Diff}_0(X)\) is contractible. In particular, every element in \(\mathrm{Diff}_0(X)\) is isotopic to the identity through diffeomorphisms in \(\mathrm{Diff}_0(X)\).
The author also proves that for any real number \(t>0\), if \(X_t\) is the compact Riemann surface of genus two determined by the equation \[ \omega^2 = z(z^2-\omega)(z^2 +tz -1), \] then \(X_t\) has an anti-holomorphic automorphism of order \(4\) but no anti-holomorphic involution unless \(t=1\).
The last result provides examples of Riemann surfaces that are isomorphic to their conjugate surfaces but cannot be defined by polynomial equations with real coefficients.
Both theorems are restatments of results that were given without proof in the author’s paper [Adv. Theory Riemann Surfaces, Proc. 1969 Stony Brook Conf., 119–130 (1971; Zbl 0218.32010)].
The proofs that the author gives use connections between his provious work and works by Maclachlan and Harvey, and Birman and Hilden.
For the entire collection see [Zbl 1182.30003].

MSC:

30F10 Compact Riemann surfaces and uniformization
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
32G13 Complex-analytic moduli problems
30F60 Teichmüller theory for Riemann surfaces

Citations:

Zbl 0218.32010
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