Abstract. We relate Fuglede-Kadison determinants to entropy of finitely-presented algebraic actions in essentially complete generality. We show that if f ∈ Mm,n(Z(Γ)) is injective as a left multiplication operator on ℓ 2 (Γ) ⊕n , then the topological entropy of the action of Γ on the dual of Z(Γ) ⊕n /Z(Γ) ⊕m f is at most the logarithm of the positive Fuglede-Kadison determinant of f, with equality if m = n. We also prove that when m = n the measure-theoretic entropy of the action of Γ on the dual of Z(Γ) ⊕n /Z(Γ) ⊕n f is the logarithm of the Fuglede-Kadison determinant of f. This work completely settles the connection between entropy of principal algebraic actions and Fuglede-Kadison determinants in the generality in which dynamical entropy is defined. Our main Theorem partially generalizes results of Li-Thom from amenable groups to sofic groups. Moreover, we show that the obvious full generalization of the Li-Thom theorem for amenable groups is false for general sofic groups. Lastly, we undertake a study of when the Yuzvinskiǐ addition formula fails for a non-amenable sofic group Γ, showing it always fails if Γ contains a nonabelian free group, and relating it to the possible values of L 2 -torsion in general.

Abstract. We define and investigate the singular subspace Hs(N ⊆ M ) of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical N -N subbimodule of L 2 (M ) containing the quasinormalizer introduced in [38], the one-sided quasinormalizer introduced in [11], and the wq-normalizer introduced in [14] (following upon work in [26] and [40]). We then obtain a weak notion of regularity (called spectral regularity) by demanding that the singular subspace of N ⊆ M generates M. By abstracting Voiculescu's original proof of absence of Cartan subalgebras in [53] we show that there can be no diffuse, hyperfinite subalgebra of L(Fn) which is spectrally regular. Our techniques are robust enough to repeat this process by transfinite induction and rule out chains of spectrally regular inclusions of algebras starting from a diffuse, hyperfinite subalgebra and ending in L(Fn). We use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of II 1 -factors (see [14]). Our results may be regarded as a consistency check for the possibility of existence of a "good" cohomology theory of II 1 -factors. We can also use our techniques to show that if Ut is a one-parameter orthogonal group on a real Hilbert space H and the spectral measure of its generator is singular with respect to the Lebesgue measure, then the continuous core of the free Araki-Woods factor Γ(H, Ut) ′′ is not isomorphic to L(Ft⊗B(ℓ 2 (N)) for any t ∈ (1, ∞]. In particular, Γ(H, Ut) ′′ ∼ = Γ(L 2 (R, m), λt) ′′ where m is Lebesgue measure and λ is the left regular representation. This was previously only know when the spectral measure of the generator of Ut had all of its convolution powers singular with respect to Lebesgue measure. We give similar applications to crossed products by free Bogoliubov actions in the spirit of [22].

We deduce properties of the Koopman representation of a positive entropy probability measurepreserving action of a countable, discrete, sofic group. Our main result may be regarded as a "representationtheoretic" version of Sinaǐ's Factor Theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if Γ is a nonamenable group and Γ (X, µ) is a probability measurepreserving action which is not strongly ergodic, then no action orbit equivalent to Γ (X, µ) has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.

Given the large class of groups already known to be sofic, there is seemingly a shortfall in results concerning their permanence properties. We address this problem for wreath products, and in particular investigate the behaviour of more general metric approximations of groups under wreath products.Our main result is the following. Suppose that H is a sofic group and G is a countable, discrete group. If G is sofic, hyperlinear, weakly sofic, or linear sofic, then G H is also sofic, hyperlinear, weakly sofic, or linear sofic respectively. In each case we construct relevant metric approximations, extending a general construction of metric approximations for G H that uses soficity of H.

Following the methods of [13], we introduce an extended version of von Neumann dimension for representations of a discrete, measure-preserving, sofic equivalence relation. Similar to [13], this dimension is decreasing under equivariant maps with dense image, and in particular is an isomorphism invariant. We compute dimensions of L p (R, µ) ⊕n for 1 ≤ p ≤ 2. We also define an analogue of the first l 2 -Betti number for l p -cohomology of equivalence relations, provided the equivalence relations satisfies a certain "finite presentation" assumption. This analogue of l 2 -Betti numbers may shed some light on the conjecture that cost (as defined by Levitt in [17]) is one more than l 2 -Betti number (as defined by Gaboriau in [9]) of equivalence relations.