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arXiv · 1403.6981

Revisiting Farrell's nonfiniteness of Nil

Jean-François Lafont·Stratos Prassidis·Kun Wang
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Abstract

We study Farrell Nil-groups associated to a finite order automorphism of a ring $R$. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite multiplicity. If the original finite group is a direct summand, then the countably infinite sum of the finite subgroup also appears as a direct summand. We use this to deduce a structure theorem for countable Farrell Nil-groups with finite exponent. Finally, as an application, we show that if $V$ is any virtually cyclic group, then the associated Farrell or Waldhausen Nil-groups can always be expressed as a countably infinite sum of copies of a finite group, provided they have finite exponent (which is always the case in dimension $0$).

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BibTeXRIS

Jean-François Lafont, Stratos Prassidis, Kun Wang. 2014-07-16. Revisiting Farrell's nonfiniteness of Nil. https://doi.org/10.2140/akt.2016.1.209

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