Benjamin Hayes: Fuglede-Kadison determinants and sofic entropy

Speaker: Benjamin Hayes

Title:  Fuglede-Kadison determinants and sofic entropy

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 (our results apply to a larger class of actions than this). 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.