| The Teichmuller modular group is known to act biholomorphically and isometrically on the Teichmuller space of Riemann surfaces, T, (equipped with the Teichmuller metric). The fix-point sets of these maps are complex submanifolds of T. Let f be an involutory element of the modular group. Since there is a unique geodesic in T joining any pair of distinct points we obtain a retraction, H(,f), of T onto the fix-point set of f by mapping any point x of T to the midpoint of the geodesic segment joining x and f(x). For example, we can thus associate naturally to any Riemann surface a hyperelliptic one.;We proceed to develop general criteria for answering the following: if g is a C('1)-submersion from a complex manifold X to a real manifold Y, then when does there exist a complex structure on Y making g holomorphic? Two sets of necessary and sufficient conditions are discovered and a generic example is given showing the essential nature of the conditions developed.;The above criteria find application in providing conditions for our retractions H(,f) to be holomorphic.;We investigate in detail one of these H(,f) maps on T(1,2),--the Teichmuller space of tori with two punctures. We define a holomorphic retraction H from T(1,2) to the fix-point set of f. The fibers of H are the fibers of the natural projection (pi): T(1,2) (--->) T(1,1) and the fibers of H(,f) are shown to be geodesic discs. This suggests a general question which we answer by proving that the fibers of the natural projection (pi): T(g, n + 1) (--->) T(g,n), ((pi) is obtained by 'forgetting a puncture'), are never geodesic discs in T(g, n + 1) unless (g,n) = (0,3). The proof comes from some of Bers' results on the non-existence of invariant lines under the action of certain ('reducible') elements of the modular group. In particular, this explicitly settles the question of embedding proper holomorphic discs into Teichmuller spaces which are not the well-understood geodesic discs.;We study H(,f), proving that it is continuous and characterizing the fibers (i.e. the sets H(,f)('-1)(x)), in terms of geodesics whose directions are determined by eigenvectors of the action of df on the tangent space to T at the fix-point x of f.;Clearly, the results above prove H(,f) (NOT=) H; this sets off in interesting contrast our theorem that H(,f) and H coincide up to first-order approximation near their common target set. This latter result leads, via the perturbation formula for solutions of Beltrami equations, to the vanishing of a certain integral involving classical elliptic functions. We understand this better by utilizing some classical function theory involving the evaluation of an interesting double-series.;The construction of H mentioned above leads to some general holomorphic covering spaces in which some explicit complex domains D(,0,n) and D(,1,)('m) are covered by the Teichmuller spaces T(0,n) and T(1,m) respectively. The domain D(,g,k) are hyperbolic complex manifolds and in fact they turn out to be the classical Torelli spaces of punctured spheres and tori. The sphere case was known to D. Patterson. In the case of the torus our result comes from a topological theorem we have characterizing those diffeomorphisms on a punctured torus whose homology action is identity.;Since the Teichmuller modular group is the full group of automorphisms of Teichmuller space (leaving out a few exceptional cases), we are able to exhibit explicitly the full holomorphic automorphism groups of the domains D(,g,k)((g,k) (NOT=) (1,2)). Indeed we can express the Torelli modular groups on D(,1,m) as groups of integer matrices. |
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