Algebraic entropy of amenable group actions
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- 10.48550/arXiv.1410.8306
- arXiv:
- arXiv:1410.8306
- Bibcode:
- 2014arXiv1410.8306V
- Keywords:
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- Mathematics - Rings and Algebras;
- Mathematics - Group Theory;
- Mathematics - Representation Theory;
- 18E15;
- 18E35;
- 16S35;
- 16D10;
- 43A07