Von Neumann Orbit Equivalence

This thesis generalizes the notion of orbit equivalence to the non-commutative setting by introducing a new equivalence relation on groups, which we call von Neumann orbit equivalence (vNOE). This thesis proves the stability of this equivalence relation under taking free products and graph products of groups. To achieve this, von Neumann orbit equivalence of tracial von Neumann algebras is introduced. This thesis shows that two countable discrete groups vNOE if and only if the corresponding group von Neumann algebras are vNOE, and that vNOE of tracial von Neumann algebras is stable under taking free products and graph products of tracial von Neumann algebras.