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Theses

Géométrie des variétés de représentations dans des groupes de Lie résolubles et géométrie en dimension 3

Abstract : In this thesis, we are interested in the character variety of a surface group with respect to a solvable Lie group.This is the the set of all homomorphisms of the fundamental group of a surface taking values in a Lie group, up to conjugacy.Character varieties have been intensively studied, mostly for the case of complex semisimple Lie groups, or more generally reductive groups.Here we focus on real solvable Lie groups.Actually, we will mostly be concerned with metabelian and nilpotent groups.In the first part, we study the topological structure of the character variety in such a setting, starting with the example of the real Heisenberg group, before settling the general case.In particular, we will study the Mapping Class Group's action on the character variety, as a first step towards drawing a parallel with the lower central series of the surface fundamental group.We explore the geometry of the character variety and representation variety in solvable and nilpotent Lie groups and show that it has a contact structure, and a natural Riemannian pseudo-metric of null signature.Bounded characteristic classes for semisimple Lie groups were largely studied in relation with the so-called Milnor-Wood inequalities.They provide bounds for the number of connected components of character varieties.In the solvable context, the number of connected components might be infinite and even for the Heisenberg group.Eventually we consider the Torelli group action on the character variety for surfaces with boundary, near points corresponding to abelian representation classes.We first show that the tangent representation factors through the abelianization.Finally, we prove that this abelian representation of the Torelli group at a generic point is equivalent to the Johnson homomorphism,if the Lie group is the Malcev completion of the second nilpotent quotient of the surface group.
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Submitted on : Friday, May 13, 2022 - 6:15:37 PM
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Clément Berat. Géométrie des variétés de représentations dans des groupes de Lie résolubles et géométrie en dimension 3. Topologie algébrique [math.AT]. Université Grenoble Alpes [2020-..], 2021. Français. ⟨NNT : 2021GRALM058⟩. ⟨tel-03667937⟩

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