The geometry of the handlebody groups I: Distortion
  We show that the mapping class group of a handlebody of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface. The same holds true for solid tori with at least two marked points or spots. 
Ursula Hamenst\"adt and Sebastian Hensel
math.GR math.GT math.MG