Title: Fuglede-Kadison determinants and sofic entropy
Speaker: Ben Hayes (Vanderbilt University)
Abstract: Let G be a countable, discrete group, an algebraic action of G is an action by automorphisms of a compact, abelian, metrizable group X. The data of an algebraic action is equivalent, via Pontryagin duality, to a countable Z(G)-module A, where Z(G) is the integral group ring. A particular case of interest is as follows: fix f in Z(G), and let X_{f} be the Pontraygin dual of Z(G)/Z(G)f (as an abelian group). This is called a principal algebraic action. There has been a long history of connecting the entropy of the action of G on X_{f} to the Fuglede-Kadison determinant (defined via the von Neumann algebra of G) of f in various degrees of generality. In the amenable case, this was studied by Lind-Schmidt-Ward, Deninger, Deninger-Schmidt, Li and completely settled by Li-Thom. We study the entropy of such actions when G is sofic (using sofic entropy as defined by Bowen, Kerr-Li). Generalizing work of Bowen, Kerr-Li, Bowen-Li (as well as the amenable case) we completely settle the connection between FugledeK adison determinants and sofic entropy of principal algebraic actions. We will comment on the techniques, which differ from the amenable case and are the first to avoid approximating the Fuglede-Kadison determinant of f by finite-dimensional determinants. No knowledge of sofic entropy, Fuglede-Kadison determinants or von Neumann algebras will be assumed.
