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We show cusps are dense in Bers' boundary for Teichm uller space. The proof rests on an estimate for the algebraic eeect of a unit quasiconformal deformation supported in the thin part of a hyperbolic Riemann surface.
Equations that can be solved using iterated rational maps are characterized: an equation is ‘computable’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermite and Kronecker) is replaced by a purely iterative(More)
Any covering Y ! X of a hyperbolic Riemann surface X of-nite area determines an inclusion of Teichm uller spaces Teich(X) ,! Teich(Y). We show this map is an isometry for the Teichm uller metric if the covering is amenable, and contracting otherwise. In particular , we establish jjjj < 1 for classical Poincar e series (Kra's `Theta conjecture'). The(More)
Let X be a hyperbolic Riemann surface or orbifold, possibly of infinite topological complexity. Let φ : X → X be a quasiconformal map. We show the following conditions are equivalent (§1): (a) φ has a lift to the universal cover ∆ which is the identity on S1; (b) φ is homotopic to the identity rel the ideal boundary of X; and (c) φ is isotopic to the(More)