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Anosov representations and Margulis spacetimes

The goal of this project is to establish a precise relationship between Margulis spacetimes and Anosov representations.

Margulis spacetimes are geometric objects obtained from proper actions of discrete groups on real three dimensional spaces. Margulis spacetimes can be alternatively be described as deformations of hyperbolic structures. One very important information of its geometry lies on the boundary.

Anosov representations are a systematic way to study representations of a discrete group via its behaviour on the boundary. So one can hope that Margulis spacetimes will lend itself being studied via Anosov representations.

The main objectives of this project can be broadly divided into four parts:

• Construct coordinates on the space $$G/P$$ using generalised cross ratios where $$P$$ is a pseudo-parabolic subgroup of $$G=SO^0(n,n+1)\ltimes{\mathbb R}^{2n+1}$$.
• Calculate the curvature and describe the geodesics of the pressure metric.
• Introduce the notion of an Anosov representation for general non-semisimple Lie groups of the form $$G\ltimes V$$ and relate it with proper actions on $$V$$ where $$G$$ is a semisimple Lie group acting irreducibly on some vector space $$V$$.
• Investigate the boundaries of the moduli space of Anosov representations via degenerations of Anosov structures.

## Publications

In this article, we interpret affine Anosov representations of any word hyperbolic group in $$\mathsf{SO}_0(n−1,n)⋉\mathbb{R}^{2n−1}$$ as infinitesimal versions of representations of word hyperbolic groups in $$\mathsf{SO}_0(n,n)$$ which are both Anosov in $$\mathsf{SO}_0(n,n)$$ with respect to the stabilizer of an oriented (n−1)-dimensional isotropic plane and Anosov in $$\mathsf{SL}(2n,\mathbb{R})$$ with respect to the stabilizer of an oriented n-dimensional plane. Moreover, we show that representations of word hyperbolic groups in $$\mathsf{SO}_0(n,n)$$ which are Anosov in $$\mathsf{SO}_0(n,n)$$ with respect to the stabilizer of an oriented (n−1)-dimensional isotropic plane, are Anosov in $$\mathsf{SL}(2n,\mathbb{R})$$ with respect to the stabilizer of an oriented n-dimensional plane if and only if its action on $$\mathsf{SO}_0(n,n)/\mathsf{SO}_0(n-1,n)$$ is proper. In the process, we also provide various different interpretations of the Margulis invariant.