Vol. 1, No. 2, 2016

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Revisiting Farrell's nonfiniteness of Nil

Jean-François Lafont, Stratos Prassidis and Kun Wang

Vol. 1 (2016), No. 2, 209–225

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 zero).

Nil-groups, algebraic $K$-theory, Frobenius functors, Verschiebung functors
Mathematical Subject Classification 2010
Primary: 18F25, 19D35, 18E10
Received: 23 February 2015
Accepted: 31 March 2015
Published: 20 October 2015
Jean-François Lafont
Department of Mathematics
The Ohio State University
100 Math Tower
231 West 18th Avenue
Columbus, OH 43210-1174
United States
Stratos Prassidis
Department of Mathematics
University of the Aegean
Karlovassi, Samos, 83200
Kun Wang
Department of Mathematics
Vanderbilt University
Nashville, TN 37240
United States