Topological finiteness and stability of hyperbolizable manifolds

In this thesis we study the topology of closed hyperbolizable manifolds with bounded diameter and bounded volume entropy. We prove that their fundamental group contains free subgroups of large rank and generators of universally bounded length. This imply an entropy-cardinality inequality, and their topological finiteness. We also prove a systole-entropy inequality which imply, in dimension 3, their topological stability. The philosophy behind these results is that we can think the volume entropy as an average, large scale version of the Ricci curvature. Hence, it allows us to prove analogues of classical results in Riemannian geometry, with a very strong topological hypothesis (being hyperbolizable), but with no assumptions at all on their curvature. The quotes at the beginning of each chapter, arguably out of context, strive to recreate, for the reader, the author’s unsettling, and emotionally tolling experience of working on a PhD in mathematics during a climate and ecological crisis. The appendix, which aims at explaining the work in accessible terms, is an attempt of resisting through community.