The study of hyperbolic manifolds often begins with the thick-thin decomposition. Given a number epsilon > 0, we decompose a manifold into the epsilon-thin part (points on essential loops of length less than epsilon), and the epsilon-thick part (everything else). The Margulis lemma says that there is a universal number epsilon_n, depending only on the dimension (and nothing else!), such that the thin part of every hyperbolic n-manifold has very simple topology.
In dimension 3, we still do not know the optimal Margulis constant epsilon_3. Part of the problem is that while the topology is simple, the geometry of epsilon-thin tubes can be quite complicated. I will describe some joint work in progress with Jessica Purcell and Saul Schleimer, which can help give quantitative control on the Margulis constant.
In dimension 3, we still do not know the optimal Margulis constant epsilon_3. Part of the problem is that while the topology is simple, the geometry of epsilon-thin tubes can be quite complicated. I will describe some joint work in progress with Jessica Purcell and Saul Schleimer, which can help give quantitative control on the Margulis constant.