Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T16:43:43.537Z Has data issue: false hasContentIssue false
  • English
  • Français

Homology of some Artin and twisted Artin Groups

Published online by Cambridge University Press:  21 September 2009

Maura Clancy  and
Graham Ellis
Maura Clancy
Affiliation:
Mathematics Department, National University of Ireland, Galway, mclancy@nuigalway.ie.
Graham Ellis
Affiliation:
Mathematics Department, National University of Ireland, Galway, graham.ellis@nuigalway.ie.
Get access

Abstract

We begin the paper with a simple formula for the second integral homology of a range of Artin groups. The formula is derived from a polytopal classifying space. We then introduce the notion of a twisted Artin group and obtain polytopal classifying spaces for a range of such groups. We demonstrate that these explicitly constructed spaces can be implemented on a computer and used in homological calculations.

Keywords

Artin groupClassifying spaceGroup cohomology
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 1 , August 2010 , pp. 171 - 196
Copyright
Copyright © ISOPP 2009

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

1.Adem, A., Ge, J., Pan, J. and Petrosian, N., “Compatible actions and cohomology of crystallographic groups”, arXiv 0704.1823v1, (2007).Google Scholar
2.Appel, K.J. and Schupp, P.E., “Artin groups and infinite Coxeter groups”, Invent. Math., 72 (1983), 201–220.Google Scholar
3.Blind, R. and Mani-Levitska, P., “Puzzles and polytope isomorphisms”, Aequationes Math., 34 (1987), 287–297.Google Scholar
4.Brady, T., “Free resolutions for semi-direct products”, Tôhoku Math. J. 45 (1993), 535–537.CrossRefGoogle Scholar
5.Brady, T. and McCammond, J., “Three-generator Artin groups of large type are biautomatic”, J. Pure Applied Algebra, 151 no. 1 (2000), 1–9.CrossRefGoogle Scholar
6.Bridson, M. and Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der math. Wissensch., vol. 319 (Springer 1999).CrossRefGoogle Scholar
7.Callegari, F., Moroni, D. and Salvetti, M., “The K(π,1) problem for the affine Artin group of type and its cohomology”, arXiv:0705.2830.Google Scholar
8.Charney, R. and Davis, M.W., “The K(π,1) problem for hyperplane complements associated to infinite reflection groups”, Journal Amer. Math. Soc., vol. 8, issue 3 (1995), 597–627.Google Scholar
9.Charney, R. and Davis, M.W., “Finite K(π,1)s for Artin groups”, Prospects in topology (Princeton, NJ, 1994), 110–124, Ann. of Math. Stud., 138, Princeton Univ. Press, Princeton, NJ, 1995.Google Scholar
10.Charney, R. and Peifer, D., “The K(π,1)-conjecture for the affine braid groups”, Comment. Math. Helv., 78 no. 3 (2003), 584–600.Google Scholar
11.Chiswell, I.M., “The cohomological dimension of 1-relator groups”, J. London Math. Soc. (2) 11 (1975) 381–382.Google Scholar
12.Coxeter, H.S.M., “Discrete groups generated by reflections”, Annals of Math. (2) 35 (1934), no. 3, 588–621.CrossRefGoogle Scholar
13.Clancy, M., “Explicit small classifying spaces for a range of finitely presented infinite groups”, PhD thesis, NUI Galway 2009.CrossRefGoogle Scholar
14.Dehornoy, P. and Lafont, Y., “Homology of Gaussian groups”, Annales Inst. Fourier (Grenoble), 53 no. 2 (2003), 489–540.CrossRefGoogle Scholar
15.Eick, B., Gähler, F. and Nickel, W., “Cryst”, a package for the GAP computer algebra system, available from http://www.gap-system.org.Google Scholar
16.Ellis, G., “HAP - Homological Algebra Programming”, a package for the GAP computer algebra system, available from http://www.gap-system.org.Google Scholar
17.Ellis, G. and Sköldberg, E., “The K(π,1)-conjecture for a class of Artin groups”, Comment. Math. Helv., to appear.Google Scholar
18.Howlett, R.B., “On the Schur multipliers of Coxeter groups”, J. London Math. Soc (2), 28 no. 2 (1988), 263–276.Google Scholar
19.Humphreys, J.E., Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics 29 (CUP, 1990).Google Scholar
20.Kapranov, M. and Saito, S., “Hidden Stasheff polytpoes in algebraic K–theory and the space of Morse functions”, Contemp. Math. 227 (1999), 191–225.Google Scholar
21.Salvetti, M., “The homotopy type of Artin groups, Math. Res. Lett., 1 no. 5 (1994), 565–577.CrossRefGoogle Scholar
22.Squier, C.C., “The homological algebra of Artin groups”, Math. Scand., 75 no. 1 (1994), 5–43.Google Scholar
23.Wall, C.T.C. “Resolutions for extensions of groups”, Proc. Cambridge Philos. Soc. 57 (1961), 251–255.CrossRefGoogle Scholar