Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1389–1417 | Cite as

Algebraic entropy of amenable group actions

  • Simone ViriliEmail author


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.


Length functions Gabriel dimension Algebraic entropy Amenable groups Zero divisors Stable finiteness 

Mathematics Subject Classification

16S35 43A07 16D10 18E35 



It is a pleasure for me to thank Peter Vámos for giving me a copy of his Ph.D. thesis and for useful discussions started in Padova in 2010. I am also sincerely grateful to my Ph.D. advisor Dolors Herbera for her encouragement and trust: she gave me time, freedom and several suggestions that helped me to work independently on this project. Finally, I would like to thank Pere Ara, Ferran Cedó, Hanfeng Li, Bingbing Liang and Nhan-Phu Chung for some useful discussions on preliminary versions of this paper.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Facultad de matemáticasUniversidad de MurciaMurciaSpain

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