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Exposant critique des groupes de surfaces agissant sur H2 x H2 et H3

Abstract : This aim of this thesis is the study of the critical exponent associated to a surface group acting on two different spaces. First we study the diagonal action of two teichmuller representations on the product of hyperbolic planes. Then we study quasi-Fuchsian action on the hyperbolic 3-space. The first chapter is dedicated to introduce the basic notions we need to understand the different theorems and proofs in the thesis. The study of critical exponent on H2*H2 is made in chapters 2 and 3. In chapter 2 we study the Manhattan curve, as defined by M. Burger, and more or less classical invariants as critical exponent, critical exponent with given slope, correlation coefficient. In chapter 3, we survey some results on geometric Teichmüller theory, as geodesic currents and earthquakes. We conclude this Chapter by the principal theorem of this first part, that is to say, an isolation result, improving a rigidity result of Bishop-Steger. In the last chapter, we study quasi-Fuchsian representations. The main result is an inequality between critical exponent and volume entropy of embedded surfaces. Moreover we precise the equality case, which gives a theorem of rigidity for the critical exponent.
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Submitted on : Tuesday, September 1, 2015 - 11:42:06 AM
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Olivier Glorieux. Exposant critique des groupes de surfaces agissant sur H2 x H2 et H3. Mathématiques générales [math.GM]. Université Pierre et Marie Curie - Paris VI, 2015. Français. ⟨NNT : 2015PA066127⟩. ⟨tel-01189128⟩



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