Effective quasimorphisms on free chains
  We find homogeneous counting quasimorphisms that are effective at seeing chains in a free group F. As corollary, we derive that if a group G has an index-d free subgroup, then every element g in G either has stable commutator length at least 1/8d or some power of g is conjugate to its inverse. We also show that for a finitely-generated free group F, there is a countable basis for the real vector space of homogeneous quasimorphisms on F. 
Jing Tao
math.GR