Anosov representations and Margulis spacetimes

• Applicant: Dr. Sourav Ghosh
• Subject Area: Mathematics
• Term: from 2017 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 338519652

This project investigates the representations of finitely generated hyperbolic groups into affine Lie groups using their behaviour at infinity.The construction and classification of proper affine actions have been an important area in mathematics which received revived attention in recent years. The famous Auslander conjecture states that any affine crystallographic group is virtually solvable. In the 1980s Margulis gave an example of a free group acting properly on the real three dimensional space as affine transformations. His example showed that the cocompactness assumption in the Auslander conjecture is crucial, thereby settling the question of its necessity raised by Milnor. It lead to the construction of so called Margulis space times. A Margulis spacetime is the quotient of a three dimensional space by a finitely generated, free, non-abelian group acting properly and freely as affine transformations. Since then there has been a growing interest in the construction of proper affine actions of finitely generated hyperbolic groups, as well as in the study of the deformation spaces of all such proper affine actions of a fixed group on a fixed affine space. In this project these questions are approached through the new theory of Anosov representations into affine Lie groups, which provide new tools, such as generalized cross ratios, the thermodynamic formalism, as well as the insights from degeneration of Anosov representations into semisimple Lie groups. Consequently, the goal is not only to prove that Anosov representations into affine Lie groups give rise to proper affine actions, but also to investigate more closely the space of Anosov reprentations.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity