Mean dimension, mean length, and von Neumann-Luck rank

In this dissertation, we focus on the connections between invariants in dynamical systems and invariants in L.;2-invariants theory.;Joint with Li, we establish a connection between the mean dimension in dynamical systems and the von Neumann-Luck rank in L.;2-invariants theory. To establish this connection, we introduce an invariant called mean length for modules of group rings. In Chapter 3, we deal with the case of amenable group actions. Thanks to the dimension-flatness of the group von Neumann algebra for amenable groups, we have the addition formula for von Neumann-Luck rank, which serves as the second crucial ingredient to prove the equality. In Chapter 4, we generalize the results in Chapter 3 to the case of sofic group actions. By introducing the method of relativization, we overcome the obstruction from the failure of the addition formula for von Neumann-Luck rank and establish an alternative addition formula. This alternative addition formula provides us with the proper tool to carry on the idea of using the mean length as a medium to relate the mean dimension with von Neumann-Luck rank.