Algebraic entropy of amenable group actions

TL;DR

This paper introduces algebraic entropy for modules over crossed product rings of amenable groups, providing a new invariant that helps address classical conjectures in group ring theory.

Contribution

It constructs an algebraic entropy invariant for modules over crossed product rings, extending additive functions and applying them to longstanding algebraic conjectures.

Findings

Defined algebraic L-entropy for modules over crossed products

Connected algebraic entropy to stable finiteness of group rings

Provided new approaches to the Zero-Divisors Conjecture

Abstract

Let $R$ be a ring, let $G$ be an amenable group and let $R\ast G$ be a crossed product. The goal of this paper is to construct, starting with a suitable additive function $L$ on the category of left modules over $R$, an additive function on a subcategory of the category of left modules over $R\ast G$, which coincides with the whole category if $L({}_RR) <\infty$. This construction can be performed using a dynamical invariant associated with the original function $L$, called algebraic $L$-entropy. We apply our results to two classical problems on group rings: the Stable Finiteness and the Zero-Divisors Conjectures.