We study Farrell Nil-groups associated to a finite-order automorphism of a ring . 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 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 zero).
"Revisiting Farrell's nonfiniteness of Nil." Ann. K-Theory 1 (2) 209 - 225, 2016. https://doi.org/10.2140/akt.2016.1.209