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Media type:
Doctoral Thesis;
Electronic Thesis;
E-Book
Title:
Generalized Schottky groups, oriented flag manifolds and proper actions
Contributor:
Treib, Nicolaus
[Author]
Published:
Heidelberg University: HeiDok, 2018
Language:
English
DOI:
https://doi.org/10.11588/heidok.00025317
Origination:
Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
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 ...