Ursula HAMENSTÄDT - The geometry of  3 - manifolds before and after Perelman
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2017-12-27

The  rank  of  a  hyperbolic  manifold  is  the  smallest  number  of  generators  of  its fundamental  group.  McMullen  conjectured  that  for  all  $k\geq  2$,  the  pointwise injectivity  radius  of  a  closed  hyperbolic  3-manifold  of  rank at  most  k  is  uniformly 
bounded from above. We explain some methods which were introduced before and after the foundational work of Perelman to study these manifolds, and we show how these  methods  can  be  used  to  prove  McMullen's  conjecture  in  many  cases  inc
luding random 3-manifolds.