Algebraic & Geometric Topology

Non-finiteness results for Nil-groups

Joachim Grunewald

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Generalizing an idea of Farrell we prove that for a ring Λ and a ring automorphism α of finite order the groups Nil0(Λ;α) and all of its p–primary subgroups are either trivial or not finitely generated as an abelian group. We also prove that if β and γ are ring automorphisms such that βγ is of finite order then Nil0(Λ;Λβ,Λγ) and all of its p–primary subgroups are either trivial or not finitely generated as an abelian group. These Nil-groups include the Nil-groups appearing in the decomposition of Ki of virtually cyclic groups for i1.

Article information

Algebr. Geom. Topol., Volume 7, Number 4 (2007), 1979-1986.

Received: 6 May 2006
Accepted: 6 September 2006
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 18F25: Algebraic $K$-theory and L-theory [See also 11Exx, 11R70, 11S70, 12- XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67]
Secondary: 19B28: $K_1$of group rings and orders [See also 57Q10] 19D35: Negative $K$-theory, NK and Nil

Nil-groups non-finiteness twisted Laurent polynomial ring


Grunewald, Joachim. Non-finiteness results for Nil-groups. Algebr. Geom. Topol. 7 (2007), no. 4, 1979--1986. doi:10.2140/agt.2007.7.1979.

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  • A Bartels, W Lück, Isomorphism conjecture for homotopy $K$–theory and groups acting on trees, J. Pure Appl. Algebra 205 (2006) 660–696
  • F T Farrell, The obstruction to fibering a manifold over a circle, Indiana Univ. Math. J. 21 (1971/1972) 315–346
  • F T Farrell, The nonfiniteness of Nil, Proc. Amer. Math. Soc. 65 (1977) 215–216
  • F T Farrell, W-C Hsiang, A formula for $K_{1}R_{\alpha}[T]$, from: “Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968)”, Amer. Math. Soc., Providence, R.I. (1970) 192–218
  • F T Farrell, L E Jones, The lower algebraic $K$–theory of virtually infinite cyclic groups, K–Theory 9 (1995) 13–30
  • D R Grayson, The $K$–theory of semilinear endomorphisms, J. Algebra 113 (1988) 358–372
  • J Grunewald, The Behavior of Nil-Groups under Localization and the Relative Assembly Map, Preprintreihe SFB 478–Geometrische Strukturen in der Mathematik, Münster, Heft 429 (2006)
  • A O Kuku, G Tang, Higher $K$–theory of group-rings of virtually infinite cyclic groups, Math. Ann. 325 (2003) 711–726
  • R Ramos, Non Finiteness of twisted Nils, preprint (2006)
  • F Waldhausen, Algebraic $K$–theory of generalized free products I, II, Ann. of Math. $(2)$ 108 (1978) 135–204
  • F Waldhausen, Algebraic $K$–theory of generalized free products III, IV, Ann. of Math. $(2)$ 108 (1978) 205–256