Games and elementary equivalence of $\rm II_1$ factors
  We use Ehrenfeucht-Fra\"iss\'e games to give a local geometric criterion for elementary equivalence of II$_1$ factors. We obtain as a corollary that two II$_1$ factors are elementarily equivalent if and only their unitary groups are elementarily equivalent as $\mathbb Z_4$-metric spaces. 
Isaac Goldbring, Thomas Sinclair
math.LO math.OA