Annals of K-Theory

On the $K$-theory of linear groups

Daniel Kasprowski

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove that for a finitely generated linear group over a field of positive characteristic the family of quotients by finite subgroups has finite asymptotic dimension. We use this to show that the K-theoretic assembly map for the family of finite subgroups is split injective for every finitely generated linear group G over a commutative ring with unit under the assumption that G admits a finite-dimensional model for the classifying space for the family of finite subgroups. Furthermore, we prove that this is the case if and only if an upper bound on the rank of the solvable subgroups of G exists.

Article information

Ann. K-Theory, Volume 1, Number 4 (2016), 441-456.

Received: 8 May 2015
Revised: 20 September 2015
Accepted: 22 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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] 19A31: $K_0$ of group rings and orders 19B28: $K_1$of group rings and orders [See also 57Q10] 19G24: $L$-theory of group rings [See also 11E81]

$K$- and $L$-theory of group rings injectivity of the assembly map linear groups


Kasprowski, Daniel. On the $K$-theory of linear groups. Ann. K-Theory 1 (2016), no. 4, 441--456. doi:10.2140/akt.2016.1.441.

Export citation


  • R. C. Alperin, “An elementary account of Selberg's lemma”, Enseign. Math. $(2)$ 33:3-4 (1987), 269–273.
  • R. C. Alperin and P. B. Shalen, “Linear groups of finite cohomological dimension”, Invent. Math. 66:1 (1982), 89–98.
  • A. Bartels and H. Reich, “Coefficients for the Farrell–Jones conjecture”, Adv. Math. 209:1 (2007), 337–362.
  • 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.
  • D. Degrijse and N. Petrosyan, “Bredon cohomological dimensions for groups acting on $\rm CAT(0)$-spaces”, Groups Geom. Dyn. 9:4 (2015), 1231–1265.
  • R. J. Flores and B. E. A. Nucinkis, “On Bredon homology of elementary amenable groups”, Proc. Amer. Math. Soc. 135:1 (2007), 5–11.
  • E. Guentner, N. Higson, and S. Weinberger, “The Novikov conjecture for linear groups”, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243–268.
  • E. Guentner, R. Tessera, and G. Yu, “A notion of geometric complexity and its application to topological rigidity”, Invent. Math. 189:2 (2012), 315–357.
  • E. Guentner, R. Tessera, and G. Yu, “Discrete groups with finite decomposition complexity”, Groups Geom. Dyn. 7:2 (2013), 377–402.
  • D. Kasprowski, On the $K$-theory of groups with finite decomposition complexity, Ph.D. thesis, Westfälische Wilhelms-Universtität, Münster, 2014, hook \posturlhook.
  • D. Kasprowski, “On the $K$-theory of groups with finite decomposition complexity”, Proc. Lond. Math. Soc. $(3)$ 110:3 (2015), 565–592.
  • D. Kasprowski, “On the $K$-theory of subgroups of virtually connected Lie groups”, Algebr. Geom. Topol. 15:6 (2015), 3467–3483.
  • W. Lück, Transformation groups and algebraic $K$-theory, Lecture Notes in Mathematics 1408, Springer, Berlin, 1989.
  • W. Lück, “The type of the classifying space for a family of subgroups”, J. Pure Appl. Algebra 149:2 (2000), 177–203.
  • B. E. A. Nucinkis, “On dimensions in Bredon homology”, Homology Homotopy Appl. 6:1 (2004), 33–47.
  • E. K. Pedersen and C. A. Weibel, “A nonconnective delooping of algebraic $K$-theory”, pp. 166–181 in Algebraic and geometric topology (New Brunswick, NJ, 1983), edited by A. Ranicki et al., Lecture Notes in Mathematics 1126, Springer, Berlin, 1985.
  • D. A. Ramras, R. Tessera, and G. Yu, “Finite decomposition complexity and the integral Novikov conjecture for higher algebraic $K$-theory”, J. Reine Angew. Math. 694 (2014), 129–178.
  • J. Roe, Lectures on coarse geometry, University Lecture Series 31, American Mathematical Society, Providence, RI, 2003.
  • A. Selberg, “On discontinuous groups in higher-dimensional symmetric spaces”, pp. 147–164 in Contributions to function theory (Bombay, 1960), edited by K. Chandrasekharan, Tata Institute of Fundamental Research, Bombay, 1960.
  • C. Wegner, “The Farrell–Jones conjecture for virtually solvable groups”, J. Topol. 8:4 (2015), 975–1016.