Independence tuples and Deninger's problem

Abstract

We define modified versions of the independence tuples for sofic entropy developed in [22]. Our first modification uses an ${\ell }^q$-distance instead of an ${\ell }^{\infty }$-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 $\{1,\dots ,d_i\},$ and forces our independence sets $J_i\subseteq \{1,\dots ,d_i\}$ to be such that ${\chi }_{J_i}-u_{d_i}(J_i)$ (i.e. the projection of ${\chi }_{J_i}$ onto mean zero functions) spans a representation of $\Gamma$ 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 $\Gamma$ is sofic, and $f\in M_n(\mathbb{Z}(\Gamma ))\cap {\mathrm{GL}}_n(L(\Gamma ))$ is not invertible in $M_n(\mathbb{Z}(\Gamma )),$ then det${}_{L(\Gamma )}(f)>1.$ This extends a consequence of the work in [15] and [22] where one needed $f\in M_n(\mathbb{Z}(\Gamma ))\cap {\mathrm{GL}}_n({\ell }^1(\Gamma )).$ As a consequence of our work, we show that if $f\in M_n(\mathbb{Z}(\Gamma ))\cap {\mathrm{GL}}_n(L(\Gamma ))$ is not invertible in $M_n(\mathbb{Z}(\Gamma ))$ then $\Gamma \curvearrowright (\mathbb{Z}(\Gamma {)}^{\oplus n}/\mathbb{Z}(\Gamma {)}^{\oplus n}f)\hat{}$ has completely positive topological entropy with respect to any sofic approximation.