Algebraic entropy of amenable group actions. (English) Zbl 1450.16019
Summary: Let \(R\) be a ring, let \(G\) be an amenable group and let \(R{*}G\) 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 \(R{*}G\), 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.
MSC:
| 16S35 | Twisted and skew group rings, crossed products |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 16D10 | General module theory in associative algebras |
| 18E35 | Localization of categories, calculus of fractions |
Keywords:
length functions; Gabriel dimension; algebraic entropy; amenable groups; zero divisors; stable finitenessReferences:
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