Isomorphism versus commensurability for a class of finitely presented   groups
  We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are first examples of such a contrastive complexity behaviour with respect to the isomorphism problem. 
Goulnara Arzhantseva, Jean-Francois Lafont and Ashot Minasyan
math.GR math.LO