Annals of K-Theory

Revisiting Farrell's nonfiniteness of Nil

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 zero).

Article information

Source
Ann. K-Theory, Volume 1, Number 2 (2016), 209-225.

Dates
Accepted: 31 March 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.akt/1510841570

Digital Object Identifier
doi:10.2140/akt.2016.1.209

Mathematical Reviews number (MathSciNet)
MR3514940

Zentralblatt MATH identifier
1330.18019

Citation

Lafont, Jean-François; Prassidis, Stratos; Wang, Kun. Revisiting Farrell's nonfiniteness of Nil. Ann. K-Theory 1 (2016), no. 2, 209--225. doi:10.2140/akt.2016.1.209. https://projecteuclid.org/euclid.akt/1510841570

References

• A. C. Bartels, “On the domain of the assembly map in algebraic $K$-theory”, Algebr. Geom. Topol. 3 (2003), 1037–1050.
• F. X. Connolly and S. Prassidis, “On the exponent of the $NK\sb 0$-groups of virtually infinite cyclic groups”, Canad. Math. Bull. 45:2 (2002), 180–195.
• J. F. Davis and W. Lück, “Spaces over a category and assembly maps in isomorphism conjectures in $K$- and $L$-theory”, $K$-Theory 15:3 (1998), 201–252.
• J. F. Davis, Q. Khan, and A. Ranicki, “Algebraic $K$-theory over the infinite dihedral group: an algebraic approach”, Algebr. Geom. Topol. 11:4 (2011), 2391–2436.
• J. F. Davis, F. Quinn, and H. Reich, “Algebraic $K$-theory over the infinite dihedral group: a controlled topology approach”, J. Topol. 4:3 (2011), 505–528.
• F. T. Farrell, “The nonfiniteness of Nil”, Proc. Amer. Math. Soc. 65:2 (1977), 215–216.
• F. T. Farrell and L. E. Jones, “Isomorphism conjectures in algebraic $K$-theory”, J. Amer. Math. Soc. 6:2 (1993), 249–297.
• F. T. Farrell and L. E. Jones, “The lower algebraic $K$-theory of virtually infinite cyclic groups”, $K$-Theory 9:1 (1995), 13–30.
• D. Grayson, “Higher algebraic $K$-theory, II (after Daniel Quillen)”, pp. 217–240 in Algebraic $K$-theory (Evanston, IL 1976), Lecture Notes in Math. 551, Springer, Berlin, 1976.
• D. R. Grayson, “The $K$-theory of semilinear endomorphisms”, J. Algebra 113:2 (1988), 358–372.
• J. Grunewald, “Non-finiteness results for nil-groups”, Algebr. Geom. Topol. 7 (2007), 1979–1986.
• J. Grunewald, “The behavior of Nil-groups under localization and the relative assembly map”, Topology 47:3 (2008), 160–202.
• J. Grunewald, J. R. Klein, and T. Macko, “Operations on the A-theoretic nil-terms”, J. Topol. 1:2 (2008), 317–341.
• T. Hüttemann, J. R. Klein, W. Vogell, F. Waldhausen, and B. Williams, “The “fundamental theorem” for the algebraic $K$-theory of spaces, I”, J. Pure Appl. Algebra 160:1 (2001), 21–52.
• W. van der Kallen, “Generators and relations in algebraic $K$-theory”, pp. 305–310 in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), edited by O. Lehto, Acad. Sci. Fennica, Helsinki, 1980.
• A. O. Kuku and G. Tang, “Higher $K$-theory of group-rings of virtually infinite cyclic groups”, Math. Ann. 325:4 (2003), 711–726.
• J.-F. Lafont and I. J. Ortiz, “Relating the Farrell Nil-groups to the Waldhausen Nil-groups”, Forum Math. 20:3 (2008), 445–455.
• J.-F. Lafont and I. J. Ortiz, “Splitting formulas for certain Waldhausen Nil-groups”, J. Lond. Math. Soc. $(2)$ 79:2 (2009), 309–322.
• A. V. Prasolov, “The nonfiniteness of the group Nil”, Mat. Zametki 32:1 (1982), 9–12. In Russian; translated in 32:1 (1982), 484–485}.
• H. Prüfer, “Untersuchungen über die Zerlegbarkeit der abzählbaren primären Abelschen Gruppen”, Math. Z. 17:1 (1923), 35–61.
• R. Ramos, “Non-finiteness of twisted nils”, Bol. Soc. Mat. Mexicana $(3)$ 13:1 (2007), 55–64.
• D. J. S. Robinson, A course in the theory of groups, Graduate Texts in Mathematics 80, Springer, New York, 1993.
• F. Waldhausen, “Algebraic $K$-theory of spaces”, pp. 318–419 in Algebraic and geometric topology (New Brunswick, NJ, 1983), edited by A. Ranicki et al., Lecture Notes in Math. 1126, Springer, Berlin, 1985.
• C. Weibel, “$NK\sb 0$ and $NK\sb 1$ of the groups $C\sb 4$ and $D\sb 4$”, Comment. Math. Helv. 84:2 (2009), 339–349.