Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T09:50:06.884Z Has data issue: false hasContentIssue false
  • English
  • Français

Twisted homological stability for extensions and automorphism groups of free nilpotent groups

Published online by Cambridge University Press:  30 June 2014

Markus Szymik
Markus Szymik*
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
Get access

Abstract

We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups. The proof presented here uses a general result that applies to arbitrary extensions of groups, and that has other applications as well.

Keywords

Homological stability19B1420F28
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 1 , August 2014 , pp. 185 - 201
Copyright
Copyright © ISOPP 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

And65.Andreadakis, S.. On the automorphisms of free groups and free nilpotent groups. Proc. London Math. Soc. 15 (1965), 239–268.CrossRefGoogle Scholar
Arn69.Arnold, V.I.. The cohomology ring of the colored braid group. Math. Not. Acad. Sci. USSR 5 (1969), 138–140.Google Scholar
Arn70.Arnold, V.I.. On some topological invariants of algebraic functions. Trans. Moscow Math. Soc. 21 (1970), 30–52.Google Scholar
Bas64.Bass, H.. K-theory and stable algebra. Inst. Hautes Études Sci. Publ. Math. No. 22 (1964), 5–60.Google Scholar
Bet02.Betley, S.. Twisted homology of symmetric groups. Proc. Amer. Math. Soc. 130 (2002), 3439–3445.Google Scholar
Bor74.Borel, A.. Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. 7 (1974), 235–272.Google Scholar
Bri73.Brieskorn, E.. Sur les groupes de tresses [d'après V.I. Arnol'd]. Séminaire Bourbaki 24ème année (1971/1972), Exp. 401, 21–44. Lecture Notes in Math 317. Springer, Berlin, 1973.Google Scholar
Cha80.Charney, R.. Homology stability for GLn of a Dedekind domain. Invent. Math. 56 (1980), 1–17.Google Scholar
CF13.Church, T., Farb, B.. Representation theory and homological stability. Adv. Math. 245 (2013), 250–314.Google Scholar
CLM72.Cohen, F.R., Lada, T.J., May, J.P.. The homology of iterated loop spaces. Lecture Notes in Mathematics 533, Springer-Verlag, Berlin, 1976.Google Scholar
Dwy80.Dwyer, W.G.. Twisted homological stability for general linear groups. Ann. of Math. 111 (1980), 239–251.Google Scholar
Hat95.Hatcher, A.. Homological stability for automorphism groups offree groups. Comment. Math. Helv. 70 (1995), 39–62.Google Scholar
HV98a.Hatcher, A., Vogtmann, K.. Cerf theory for graphs. J. London Math. Soc. 58 (1998)čň 633–655.Google Scholar
HV98b.Hatcher, A., Vogtmann, K.. Rational homology of Aut(Fn). Math. Res. Lett. 5 (1998), 759–780.Google Scholar
HW10.Hatcher, A., Wahl, N.. Stabilization for mapping class groups of 3-manifolds. Duke Math. J. 155 (2010), 205–269.CrossRefGoogle Scholar
Hop42.Hopf, H.. Fundamentalgruppe und zweite Bettische Gruppe. Comment. Math. Helv. 14 (1942), 257–309.Google Scholar
Joh83.Johnson, D.. A survey of the Torelli group. Contemp. Math. 20 (1983), 165–179.Google Scholar
vdK80.van der Kallen, W.. Homology stability for linear groups. Invent. Math. 60 (1980), 269–295.Google Scholar
Kas03.Kassabov, M.D.. On the automorphism tower of free nilpotent groups. Thesis, Yale University, 2003.Google Scholar
LS75.Lam, T.Y., Siu, M.K.. K0 and K1—an introduction to algebraic K-theory. Amer. Math. Monthly 82 (1975), 329–364.Google Scholar
Mac95.Macdonald, I.G.. Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.Google Scholar
Mag32.Magnus, W.. Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring. Math. Ann. 111 (1935), 259–280.Google Scholar
Mor93.Morita, S.. The extension of Johnson's homomorphism from the Torelli group to the mapping class group. Invent. Math. 111 (1993), 197–224.CrossRefGoogle Scholar
Nak60.Nakaoka, M.. Decomposition theorem for homology groups of symmetric groups. Ann. of Math. 71 (1960), 16–42.Google Scholar
Pir00a.Pirashvili, T.. Dold-Kan type theorem for Γ-groups. Math. Ann. 318 (2000), 277–298.Google Scholar
Pir00b.Pirashvili, T.. Hodge decomposition for higher order Hochschild homology. Ann. Sci. École Norm. Sup. 33 (2000), 151–179.CrossRefGoogle Scholar
Ser06.Serre, J.-P.. Lie algebras and Lie groups. Lectures given at Harvard University 1964. Corrected fifth printing of the second (1992) edition. Lecture Notes in Mathematics 1500, Springer-Verlag, Berlin, 2006.Google Scholar
Sus82.Suslin, A.A.. Stability in algebraic K-theory. Algebraic K-theory, Part I (Oberwolfach, 1980) 304–333. Lecture Notes in Math. 966, Springer, Berlin, 1982.Google Scholar
Ver99.Vershinin, V.V.. Braid groups and loop spaces. Russian Math. Surveys 54 (1999), 273–350.Google Scholar
Ver06.Vershinin, V.V.. Braids, their properties and generalizations. Handbook of algebra 4, 427–465, Handb. Algebr. 4, Elsevier/North-Holland, Amsterdam, 2006.Google Scholar
Wag76.Wagoner, J.B.. Stability for homology of the general linear group of a local ring. Topology 15 (1976), 417–423.Google Scholar
Wag77.Wagoner, J.B.. Equivalence of algebraic K-theories. J. Pure Appl. Algebra 11 (1977), 245–269.Google Scholar
Wit56.Witt, E.. Die Unterringe der freien Lieschen Ringe. Math. Z. 64 (1956), 195–216.Google Scholar