Codimension one subgroups and deforming hyperbolic manifolds

In this project, we study a particular type of manifolds (i.e. spaces which locally look like the three- dimensional space surrounding us, except that in our case the dimension may be greater than three). Specifically, we are interested in negatively curved manifolds and ask when such a manifold admits a metric of constant negative curvature. As an analogy one may think of the surface of a planet, which is usually quite uneven. Despite the unevenness, a planets surface can be deformed to a round sphere, which has constant (positive) curvature. This project is about the question when a space can be deformed to a very uniform (like the sphere) negatively curved space. We attempt to study this question by studying the geometry of lower-dimensional subspaces and the geometric behaviour `complementary` to these subspaces. This requires developing several geometric tools and raises a variety of questions in geometry and topology that we intend to explore. Notably, this approach enables us to think about negatively curved spaces in a way that provides new insights into known examples.

The main subject of this project was the investigation of negatively curved spaces. These arise both in mathematics and in other subjects (e.g. computer science and biology), which is why we wish to unterstand what kinds of negatively curved spaces exist. The geometric properties of negative curvature allow a fairly general study of these spaces with powerful results. In this project, we focused on spaces that do not admit a symmetric structure. In contrast to symmetric spaces, these are barely understood. Concretely, we studied how negatively curved spaces are determined by their boundary at infinity. We showed that, if this boundary is a sphere, we can fill the boundary with an aspherical manifold (a space with no singularities). Furthermore, we studied Gromov-Thurston manifolds. These arise as branched coverings over a base space. We developed a systematic theory for the entropy of branched coverings and showed that the asymptotic behaviour of entropy depends only on the geometry of the base space. An important topic in the context of negatively curved spaces are cubulations. Cubulations are decompositions of a space into cubes (of any dimension). Cubulations are a powerful tool, which is why they are even studied for their own sake. In this project, we studied isometries of cubulated spaces and showed that these frequently form a compactly generated group. This enables a series of explicit constructions, with which the existence of unusual/unknown cubulations can be studied.