Non-amenable finitely presented torsion-by-cyclic groups
  We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann's problem. Our group is an extension of a group of finite exponent n >> 1 by a cyclic group, so it satisfies the identity [x,y]^n = 1. 
A.Yu. Olshanskii, M.V. Sapir