## Mid-Term Report Summary - GEOMETRICSTRUCTURES (Deformation Spaces of Geometric Structures)

In the project “Deformation Spaces of Geometric Structures” we investigate a variety of different geometric structures. The models for the geometric structures are homogeneous spaces G/H, where G is a Lie group and H is a closed subgroup. Classical examples include the hyperbolic space, real or complex projective spaces, other examples of physical relevance are semi-Riemannian spaces of constant curvature, or Anti-de-Sitter space.

Given a fixed topological manifold M, we investigate if M can be locally endowed with a geometric structure G/H, and if it can, how many different ways we have to put the geometric structure G/H on M. The deformation spaces describe all these different ways. One way to find manifolds with the geometric structure G/H is to find a discrete subgroup L in G that acts properly on G/H.

We showed that a class of dynamically defined representations, so called Anosov representations, are closely linked to giving discrete groups which act properly on homogeneous spaces G/H. The quotient manifolds M are then manifolds with a geometric structure modeled on G/H. In general these manifolds are not compact. We defined new compactifications of M. These compactifications admit a natural stratification. They often arise from embedding the geometric structures G/H into bigger geometric structures G’/H’. Viewing M as endowed with a G’/H’ structure, we often increase the dimension of the deformation space.

Several new and interesting questions arise from viewing geometric structures G/H as subgeometries of “bigger” geometric structures G’/H’, in particular if G/H and G’/H’ have the same dimensions. For example, we can often find one parameter families of embeddings of G/H into G’/H’ and consider their limit geometries. In the limit new geometries arise.

A classical example of this is the following: Euclidean, spherical and hyperbolic geometry can be seen as subgeometries of real projective geometry. Scaling the curvature of the sphere, we get a one parameter family of spherical geometry as a subgeometry of projective geometry, which in the limit converges to Euclidean geometry. Similarly, scaling the curvature of the hyperbolic space, we get a one-parameter family of hyperbolic geometry as a subgeometry of projective geometry limiting to Euclidean geometry as well. This way we can construct transitions between hyperbolic and spherical structures through Euclidean structures. In this project we developed a general framework to describe such transitions of geometries for very general geometric structures, and determine all possible limit geometries. For example for the n-dimensional (semi-Riemannian) hyperbolic geometries, viewed as a subgeometries of projective geometry, we determine all possible limits and describe their geometric properties. Euclidean geometry is just the simplest (most degenerate) such limit.

Bringing the study of proper actions with the study of limits of geometries together raises many interesting questions, which we will address in the future.

For example, given a family of (Anosov) representations which induce a proper action on G/H, which in the limit induce an action on a limit geometry of G/H, one would like to describe necessary and sufficient conditions for the limiting action to be proper as well. This is a highly non-trivial problem, which relates - in special cases - to the challenge of finding proper affine actions.

Given a fixed topological manifold M, we investigate if M can be locally endowed with a geometric structure G/H, and if it can, how many different ways we have to put the geometric structure G/H on M. The deformation spaces describe all these different ways. One way to find manifolds with the geometric structure G/H is to find a discrete subgroup L in G that acts properly on G/H.

We showed that a class of dynamically defined representations, so called Anosov representations, are closely linked to giving discrete groups which act properly on homogeneous spaces G/H. The quotient manifolds M are then manifolds with a geometric structure modeled on G/H. In general these manifolds are not compact. We defined new compactifications of M. These compactifications admit a natural stratification. They often arise from embedding the geometric structures G/H into bigger geometric structures G’/H’. Viewing M as endowed with a G’/H’ structure, we often increase the dimension of the deformation space.

Several new and interesting questions arise from viewing geometric structures G/H as subgeometries of “bigger” geometric structures G’/H’, in particular if G/H and G’/H’ have the same dimensions. For example, we can often find one parameter families of embeddings of G/H into G’/H’ and consider their limit geometries. In the limit new geometries arise.

A classical example of this is the following: Euclidean, spherical and hyperbolic geometry can be seen as subgeometries of real projective geometry. Scaling the curvature of the sphere, we get a one parameter family of spherical geometry as a subgeometry of projective geometry, which in the limit converges to Euclidean geometry. Similarly, scaling the curvature of the hyperbolic space, we get a one-parameter family of hyperbolic geometry as a subgeometry of projective geometry limiting to Euclidean geometry as well. This way we can construct transitions between hyperbolic and spherical structures through Euclidean structures. In this project we developed a general framework to describe such transitions of geometries for very general geometric structures, and determine all possible limit geometries. For example for the n-dimensional (semi-Riemannian) hyperbolic geometries, viewed as a subgeometries of projective geometry, we determine all possible limits and describe their geometric properties. Euclidean geometry is just the simplest (most degenerate) such limit.

Bringing the study of proper actions with the study of limits of geometries together raises many interesting questions, which we will address in the future.

For example, given a family of (Anosov) representations which induce a proper action on G/H, which in the limit induce an action on a limit geometry of G/H, one would like to describe necessary and sufficient conditions for the limiting action to be proper as well. This is a highly non-trivial problem, which relates - in special cases - to the challenge of finding proper affine actions.