The geometry Anosov subgroups in Lie groups

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The field of locally homogeneous geometric structures on manifolds generalizes the field of cartography, in the following sense : drawing an atlas of the world is like trying to describe the shape of the earth using flat geometry, and a locally homogeneous geometric structure is like trying to describe the shape of a more general world (or manifold) using some homogeneous geometry. One of the first theorems in the subject states that it is in fact impossible to construct a complete atlas of the world without distortion, that is, that there exists no Euclidean (flat) structure on the two-dimensional sphere. Already in the study of flat structures, the Bieberbach theorems exemplify the tight relationship between geometric structures and groups of symmetries. These theorems imply that, up to finite covers, the only finite-area flat structures in 2 dimensions are similar to the world of Pac-man, that is, they are obtained by identifying parallel edges of a parallelogram. It is no coincidence that the parallelogram can tile the plane by translating multiple copies of itself, creating a tiling with translational symmetry. The study of symmetries between objects or concepts is foundational to a range of topics in pure and applied mathematics and in other sciences. We say that an object has continuous symmetry if its symmetries can be very small motions, for instance a very small rotation of a sphere is a symmetry, so it has continuous symmetry. We say that it has discrete symmetry if this is not the case, for instance a cube has discrete symmetry because it only has 90 degree rotational symmetry. The symmetries of the cube are a subset of the symmetries of the sphere. Similarly, the plane has continuous symmetry but a parallelogram tiling has discrete symmetry within the symmetries of the plane. My field lies at the intersection between the study of geometric structures and the study of discrete subgroups of continuous groups, which generalize the symmetry discussion above. I will investigate a range of examples which are generalizations of the above and work towards solutions to classification problems in those cases. These examples can then be used to formulate general conjectures and theorems. Some of the examples I am interested in have the local geometry of Einstein's static universe, a toy model of spacetime in 3 dimensions. Some more examples have local projective geometry, the type of geometry used in 3D computer graphics. Having a better understanding of these more general notions of geometry and symmetry provides important insight into the fundamental relationship between the two.