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Mean dimension and von Neumann-Lueck rank

Banff International Research Station for Mathematical Innovation and Discovery

Given a module M of the integral group ring ZG of a discrete group G, one has the von Neumann-Lueck rank of M defined in the L2-invariants theory, closely related to the l2-Betti numbers. One also has the algebraic action of G on the Pontryagin dual of M defined, which makes the dynamical invariants available. I will discuss that the von Neumann-Lueck rank corresponds to the mean dimension in the case G is sofic. Two ingredients are needed. The first is that we have to introduce the mean dimension of one action relative to an extension, which is the same as the absolute mean dimension for amenable group actions but new for sofic group actions. The second is a way to define mean length for modules of the group ring for sofic group G so that an addition formula holds, which also has application to the direct finiteness problem for group rings of G. This is joint work with Bingbing Liang.

Mathematics; Dynamical systems and ergodic theory; Global analysis, analysis on manifolds; Dynamical systems

video/mp4

Author affiliation: SUNY at Buffalo

2018-01-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64458

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/