Abstract
Algebraic K-theory is a branch of algebra dealing with linear algebra over a general ring R instead of a field. It associates to any ring R a sequence of abelian groups K n (R). The first two of these groups, K 0 and K 1, are easy to describe in concrete terms. For instance, a finitely generated projective R-module defines an element of K 0(R), and an invertible matrix over R has a “determinant” in K 1(R). The entire sequence of groups K n (R) behaves something like a homology theory of rings.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Alves, P. Ontaneda, A formula for the Whitehead group of a three-dimensional crystallographic group. Topology 45(1), 1–25 (2006)
H. Bass, K-theory and stable algebra. Inst. Hautes ’Etudes Sci. Publ. Math. 22, 5–60 (1964)
H. Bass, M. Murthy, Grothendieck groups and Picard groups of abelian group rings. Ann. Math. 86(2), 16–73 (1967)
E. Berkove, D.J. Pineda, K. Pearson, The lower algebraic K-theory of Fuchsian groups. Comment. Math. Helv. 76(2), 339–352 (2001)
E. Berkove, F.T. Farrell, D.J. Pineda, K. Pearson, The Farrell-Jones isomorphism conjecture for finite covolume hyperbolic actions and the algebraic K-theory of Bianchi groups. Trans. Am. Math. Soc. 352(12), 5689–5702 (2000)
E. Berkove, D.J. Pineda, Q. Lu, Algebraic K-theory of mapping class groups. K Theory 32(1), 83–100 (2004)
B. Bürgisser, On the projective class group of arithmetic groups. Math. Z. 184(3), 339–357 (1983)
F. Connolly, T. Koźniewski, Rigidity and crystallographic groups, I. Invent. Math. 99(1), 25–48 (1990)
D. Farley, Constructions of \(E_{\mathcal{V}\mathcal{C}}\) and \(E_{\mathcal{F}\mathcal{B}\mathcal{C}}\) for groups acting on CAT(0) spaces. Algebr. Geom. Topol. 10(4), 2229–2250 (2010)
F.T. Farrell, L. Jones, Isomorphism conjectures in algebraic K-theory. J. Am. Math. Soc. 6(2), 249–297 (1993)
F.T. Farrell, L. Jones, The lower algebraic K-theory of virtually infinite cyclic groups. K Theory 9, 13–30 (1995)
F.T. Farrell, L. Jones, Algebraic K-theory of discrete subgroups of Lie groups. Proc. Nat. Acad. Sci. USA 84(10), 3095–3096 (1987)
F.T. Farrell, W.C. Hsiang, The whitehead group of poly-(finite or cyclic) groups. J. Lond. Math. Soc. 24(2), 308–324 (1961)
F.T. Farrell, W.C. Hsiang, Manifolds with π 1 = G × α T. Am. J. Math. 95(2), 813–848 (1973)
F.T. Farrell, W.C. Hsiang, A formula for K 1(R α [T]), in Applications of Categorical Algebra. Proceedings of Symposia in Pure Mathematics, vol. 17 (American Mathematical Society, Providence, 1970), pp. 192–218
J.-F. Lafont, B. Magurn, I.J. Ortiz, Lower algebraic K-theory of certain reflection groups. Math. Proc. Camb. Philos. Soc. 148(2), 193–226 (2010)
J.-F. Lafont, I.J. Ortiz, Lower algebraic K-theory of hyperbolic 3-simplex reflection groups. Comment. Math. Helv. 84, 297–337 (2009)
W. Lück, R. Stamm, Computations of K- and L-theory of cocompact planar groups. K Theory 21(3), 249–292 (2000)
W. Lück, K- and L-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by \(\mathbb{Z}/4\). Geom. Topol. 9, 1639–1676 (2005)
K. Pearson, Algebraic K-theory of two-dimensional crystallographic groups. K Theory 14(3), 265–280 (1998)
S.P. Plotnick, Vanishing of Whitehead groups for Seirfert manifolds with infinite fundamental group. Comment. Math. Helv. 55, 654–667 (1980)
F. Quinn, Ends of maps II. Invent. Math. 68, 353–424 (1982)
J. Ratcliffe, Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149 (Springer, New York, 1994)
R.L.E. Schwarzenberger, N-Dimensional Crystallography. Research Notes in Mathematics, vol. 41 (Pittman Advanced Publishing Program, Boston, Mass.-London 1980)
G. Tsapogas, On the K-theory of crystallographic groups. Trans. Am. Math. Soc. 347(8), 2781–2794 (1995)
F. Waldhausen, Algebraic K-theory of generalized free products I, II. Ann. Math. 108, 135–256 (1978)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Farley, D.S., Ortiz, I.J. (2014). Introduction. In: Algebraic K-theory of Crystallographic Groups. Lecture Notes in Mathematics, vol 2113. Springer, Cham. https://doi.org/10.1007/978-3-319-08153-3_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-08153-3_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08152-6
Online ISBN: 978-3-319-08153-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)