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Generalized Schottky groups, oriented flag manifolds and proper actions

Treib, Nicolaus

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Abstract

This work is dedicated to the study of proper actions of discrete subgroups of Lie groups on subsets of associated homogeneous spaces.

In the first part, we study actions of discrete subgroups Γ of semi–simple Lie groups G on associated oriented flag manifolds. These are quotients G/P , where the subgroup P lies between a parabolic subgroup and its identity component. Our main result is a description of (cocompact) domains of discontinuity for Anosov representations, in terms of combinatorial data. This generalizes results of Kapovich–Leeb–Porti to the oriented setting. We show that this generalization gives rise to new domains of discontinuity that are not lifts of known ones, e.g. in the case of Hitchin representations acting on oriented Grassmannians. We also apply the finer information inherent to the oriented setup to distinguish some connected components of Anosov representations. This part constitutes joint work with Florian Stecker.

The second part of this thesis, consisting of two chapters, focuses on a method of generalizing classical Schottky groups in PSL(2,R) using partial cyclic orders. We investigate two families of spaces carrying partial cyclic orders, namely Shilov boundaries of Hermitian symmetric spaces and complete oriented flags in R^n , and prove that they both satisfy a number of topological properties. These spaces are then used to construct generalized Schottky representations into Hermitian Lie groups and PSL(n,R). We show that in the first case, generalized Schottky representations coincide with maximal representations (for surfaces with boundary) and that they yield examples of Anosov representations in both cases. Several of the results in this part are joint work with Jean–Philippe Burelle. The description of the partial cyclic order on Shilov boundaries, the relation of generalized Schottky representations in Hermitian Lie groups with maximal representations, and the analysis of generalized Schottky groups in Sp(2n,R) appeared in [BT17]. The definition of the partial cyclic order on complete oriented flags is based on discussions during a visit of Jean-Philippe to Heidelberg in September 2016.

The final part of this thesis is concerned with discrete subgroups of the group of invertible affine transformations of R^n . Let ρ : Γ → SO_0(n+1,n) \ltimes R^(2n+1) be a representation of a word hyperbolic group whose linear part is Anosov with respect to the stabilizer of a maximal isotropic subspace. We prove that properness of the induced affine action is equivalent to the nonvanishing of a generalized version of the Margulis invariant. This generalizes a theorem of Goldman–Labourie–Margulis, who proved this equivalence for representations of surface groups with Fuchsian linear parts. Our results on affine actions are joint work with Sourav Ghosh and are available on arXiv [GT17].

Item Type: Dissertation
Supervisor: Wienhard, Professor Anna
Date of thesis defense: 6 August 2018
Date Deposited: 24 Sep 2018 09:08
Date: 2018
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Mathematics
Subjects: 510 Mathematics
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