The Finitary Andrews-Curtis Conjecture
  The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent importance for computational group theory. It also resolves a question asked in [5] and shows that a computation in finite groups cannot lead to a counterexample to the classical conjecture, as suggested in [5]. 
Alexandre V. Borovik, Alexander Lubotzky and Alexei G. Myasnikov
math.GR