Independence tuples and Deninger's problem

  • Ben Hayes

    Vanderbilt University, Nashville, USA

Abstract

We define modified versions of the independence tuples for sofic entropy developed in [22]. Our first modification uses an -distance instead of an -distance. It turns out this produces the same version of independence tuples (but for nontrivial reasons), and this allows one added flexibility. Our second modification considers the „action" a sofic approximation gives on and forces our independence sets to be such that (i.e. the projection of onto mean zero functions) spans a representation of weakly contained in the left regular representation. This modification is motivated by the results in [17]. Using both of these modified versions of independence tuples we prove that if is sofic, and is not invertible in then det This extends a consequence of the work in [15] and [22] where one needed As a consequence of our work, we show that if is not invertible in then has completely positive topological entropy with respect to any sofic approximation.

Cite this article

Ben Hayes, Independence tuples and Deninger's problem. Groups Geom. Dyn. 11 (2017), no. 1, pp. 245–289

DOI 10.4171/GGD/396