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.