Algebraic & Geometric Topology

Homology and finiteness properties of $\mathrm{SL}_2(\mathbb{Z}[t,t^{-1}])$

Kevin P Knudson

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Abstract

We show that the group H2(SL2([t,t1]);) is not finitely generated, answering a question mentioned by Bux and Wortman in [Algebr. Geom. Topol. 6 (2006) 839-852].

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2253-2261.

Dates
Received: 3 September 2008
Revised: 10 November 2008
Accepted: 13 November 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796933

Digital Object Identifier
doi:10.2140/agt.2008.8.2253

Mathematical Reviews number (MathSciNet)
MR2465740

Zentralblatt MATH identifier
1167.20026

Subjects
Primary: 20F05: Generators, relations, and presentations
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Keywords
finite presentability property $\mathrm{FP}_2$ linear groups over polynomial rings

Citation

Knudson, Kevin P. Homology and finiteness properties of $\mathrm{SL}_2(\mathbb{Z}[t,t^{-1}])$. Algebr. Geom. Topol. 8 (2008), no. 4, 2253--2261. doi:10.2140/agt.2008.8.2253. https://projecteuclid.org/euclid.agt/1513796933


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References

  • P Abramenko, On finite and elementary generation of $\SL_2(\R)$
  • K-U Bux, K Wortman, A geometric proof that ${\rm SL}_2({\mathbb Z}[t,t^{-1}])$ is not finitely presented, Algebr. Geom. Topol. 6 (2006) 839–852
  • K P Knudson, The homology of ${\rm SL}_2(F[t,t^{-1}])$, J. Algebra 180 (1996) 87–101
  • K P Knudson, Unstable homotopy invariance and the homology of ${\rm SL}_2(\mathbf{Z}[t])$, J. Pure Appl. Algebra 148 (2000) 255–266
  • S Krstić, J McCool, The non-finite presentability of ${\rm IA}(F\sb 3)$ and ${\rm GL}\sb 2({\bf Z}[t,t\sp {-1}])$, Invent. Math. 129 (1997) 595–606
  • A Kurosch, Die Untergruppen der freien Produkte von beliebigen Gruppen, Math. Ann. 109 (1934) 647–660
  • J-P Serre, Trees, Springer, Berlin (1980) Translated from the French by J Stillwell
  • U Stuhler, Homological properties of certain arithmetic groups in the function field case, Invent. Math. 57 (1980) 263–281
  • A A Suslin, On the structure of the special linear group over polynomial rings, Math. USSR Izv. 11 (1977) 221–238