Abstract. We develop the theory of maximal representations of the fundamental group π1(Σ) of a compact connected oriented surface Σ with boundary ∂Σ, into the isometry group of a Hermitian symmetric space X or, more generally, a group of Hermitian type G. For any homomorphism ρ: π1(Σ) → G, we define the Toledo invariant T(Σ, ρ), a numerical invariant which is in general not a characteristic number, but which has both topological and analytical interpretations: we establish important properties, among which uniform boundedness on the representation variety Hom � π1(Σ), G � , additivity under connected sum of surfaces and congruence relations mod Z. We thus obtain information about the representation variety as well as striking geometric properties of the maximal representations, that is representations whose Toledo invariant achieves the maximum value: we show that maximal representations have discrete image, are faithful and completely reducible and they always preserve a maximal tube type subdomain of X. This extends to the case of a general Hermitian group some of the properties of the representations in

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We consider the moduli spaces of representations of the fundamental group of a surface of genus g ? 2 in the Lie groups SU(2; 2) and Sp(4; R) . It is well known that there is a characteristic number, d , of such a representation, satisfying the inequality jdj 6 2g \Gamma 2 . This allows one to write the moduli space as a union of subspaces indexed by d , each of which is a union of connected components. The main result of this paper is that the subspaces corresponding to d = \Sigma(2g \Gamma 2) are connected in the case of representations in SU(2; 2) , while they break up into 2 2g+1 + 2g \Gamma 3 connected components in the case of representations in Sp(4; R) . We obtain our results using the interpretation of the moduli space of representations as a moduli space of Higgs bundles, and an important step is an identification of certain subspaces as moduli spaces of stable triples, as studied by Bradlow and Garc'ia-Prada. 1 Introduction Let \Sigma be a closed Riemann surface of genus...

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Γg be the fundamental group of a compact orientable surface of genus g ≥ 2. We survey the study of maximal representations of Γg into G, that is the subset of Hom(Γg,G) characterized by the maximality of the Toledo invariant ([17] and [15]). Then we concentrate on the particular case G =Sp(2n, R), and we show that if ρ is any maximal representation then the image ρ(Γg) is a discrete, faithful realizations of Γg as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.

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In our previous paper [22] we proved the Bloch conjecture on rationality of secondary characteristic classes of flat rank two vector bundles over a compact Kähler manifold, and announced the full higher dimensional generalization, which we now state. 1.1 Theorem. (the Bloch conjecture) Let X be a smooth complex projective

Abstract. We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This

In this paper we prove the conjecture of Jankins and Neumann [JN2] about rotation numbers of products of circle homeomorphisms, which together with other results of [EHN] and [JN2] (mentioned below) implies that a Seifert manifold admits foliations transverse to its bers only if it admits such foliations with a projective transverse structure.

Abstract We study groups of C 1 orientation–preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly orderable groups. We show that every such group with a bounded orbit is circularly orderable, and show that certain generalized braid groups are circularly orderable. We also show that the Euler class of C ∞ diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus> 1 admits a C ∞ action with arbitrary Euler class. On the other hand, we show that Z ⊕ Z actions satisfy a homological rigidity property: every orientation– preserving C 1 action of Z ⊕ Z on the plane has trivial Euler class. This gives the complete homological classification of surface group actions on R 2 in every degree of smoothness. AMS Classification 37C85; 37E30, 57M60

1.1. Affine spaces 4 1.2. The hierarchy of structures 6