Extended von Neumann Dimension For Representations of Groups and Equivalence Relations

This thesis is on two related research problems, and is divided into 2 parts:;Part 1: Let Gamma be a countable discrete sofic group, we given an entropic formula for the von Neumann dimension of a Hilbert space representation of Gamma contained in a multiple of the left regular representation. We use our formula to extend von Neumann to any uniformly bounded representation of Gamma on a separable Banach space. We give computations for the left regular representable representation of Gamma on ℓp, as well actions on noncommutative Lp-spaces and ℓp-Betti numbers of free groups. We prove some general results about the properties of this invariant, including that the extended von Neumann dimension is always zero when the group is infinite and the representation is finite-dimensional.;Part 2: We work on an analogous problem for representations of a sofic, discrete, measure-preserving equivalence relation. Again, we are able to find an entropic formula for von Neumann dimension of a Hilbert space representation of a sofic, discrete, measure-preserving equivalence relation R. Again, this allows us to extend von Neumann dimension to actions of R on a Banach space. Following techniques of Gaboriau in [11], we are able to define the Lp-Betti numbers of (finitely presented) equivalence relations. We also indicate how this gives a potential way to solve the cost versus L 2-Betti number problem as posed by Gaboriau.