Independence tuples and Deninger's problem

Ben Hayes¹ 

¹Vanderbilt University, Nashville, USA

Groups, geometry, and dynamics, Tome 11 (2017) no. 1, pp. 245-289

Résumé

We define modified versions of the independence tuples for sofic entropy developed in [22]. Our first modification uses an lq-distance instead of an l∞-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,...,di​}, and forces our independence sets Ji​⊆{1,...,di​} to be such that χJi​​−udi​​(Ji​) (i.e. the projection of χJi​​ onto mean zero functions) spans a representation of Γ 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 Γ is sofic, and f∈Mn​(Z(Γ))∩GLn​(L(Γ)) is not invertible in Mn​(Z(Γ)), then detL(Γ)​(f)>1. This extends a consequence of the work in [15] and [22] where one needed f∈Mn​(Z(Γ))∩GLn​(l1(Γ)). As a consequence of our work, we show that if f∈Mn​(Z(Γ))∩GLn​(L(Γ)) is not invertible in Mn​(Z(Γ)) then Γ↷(Z(Γ)⊕n/Z(Γ)⊕nf)A has completely positive topological entropy with respect to any sofic approximation.

Détail

DOI : 10.4171/ggd/396

Classification : 37-XX, 22-XX, 47-XX
Mots-clés : Sofic groups, independence tuples, completely positive entropy, Fuglede–Kadison determinants