Algorithmically complex residually finite groups
  We construct the first examples of an algorithmically complex finitely presented residually finite groups and first examples of finitely presented residually finite groups with arbitrarily large (recursive) Dehn function and depth function. The groups are solvable of class 3. We also prove that the universal theory of finite solvable of class 3 groups is undecidable. 
O. Kharlampovich, A. Myasnikov, M. Sapir
math.GR math.LO