Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.

Let X be an algebraic variety over C. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over C, every holomorphic map S an → X an is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine curve C over C, every holomorphic map C an → X an is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.

Abstract. We prove a rigidity theorem for semi-arithmetic Fuchsian groups: If Γ 1 , Γ 2 are two semi-arithmetic lattices in PSL(2, R) virtually admitting modular embeddings and f : Γ 1 → Γ 2 is a group isomorphism that respects the notion of congruence subgroups, then f is induced by an inner automorphism of PGL(2, R).

Let F be a field of characteristic 0 containing all roots of unity. We construct a functorial compact Hausdorff space X F whose profinite fundamental group agrees with the absolute Galois group of F , i.e. the category of finite covering spaces of X F is equivalent to the category of finite extensions of F .The construction is based on the ring of rational Witt vectors of F . In the case of the cyclotomic extension of Q, the classical fundamental group of X F is a (proper) dense subgroup of the absolute Galois group of F . We also discuss a variant of this construction when the field is not required to contain all roots of unity, in which case there are natural Frobenius-type automorphisms which encode the descent along the cyclotomic extension.

We make some observations concerning the Galois actions on Hurwitz curves and on the closely related but lesser-known Hurwitz origamis.