Relative completions of linear groups over ℤ[𝕥] and ℤ[𝕥,𝕥⁻¹]

Knudson, Kevin

doi:10.1090/S0002-9947-99-02433-2

Relative completions of linear groups over $\mathbb {Z}[t]$ and $\mathbb {Z}[t,t^{-1}]$
HTML articles powered by AMS MathViewer

by Kevin P. Knudson

Trans. Amer. Math. Soc. 352 (2000), 2205-2216

DOI: https://doi.org/10.1090/S0002-9947-99-02433-2

Published electronically: July 26, 1999

PDF | Request permission

Correction: Trans. Amer. Math. Soc. 353 (2001), 3833-3834.

Abstract:

We compute the completion of the groups $SL_n({\mathbb Z}[t])$ and $SL_n({\mathbb Z}[t,t^{-1}])$ relative to the obvious homomorphisms to $SL_n({\mathbb Q})$; this is a generalization of the classical Malcev completion. We also make partial computations of the rational second cohomology of these groups.

References

Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Armand Borel, Stable real cohomology of arithmetic groups. II, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 21–55. MR 642850
A. K. Bousfield, Homological localization towers for groups and $\Pi$-modules, Mem. Amer. Math. Soc. 10 (1977), no. 186, vii+68. MR 447375, DOI 10.1090/memo/0186
Fritz Grunewald, Jens Mennicke, and Leonid Vaserstein, On the groups $\textrm {SL}_2(\textbf {Z}[x])$ and $\textrm {SL}_2(k[x,y])$, Israel J. Math. 86 (1994), no. 1-3, 157–193. MR 1276133, DOI 10.1007/BF02773676
Richard M. Hain, Algebraic cycles and extensions of variations of mixed Hodge structure, Complex geometry and Lie theory (Sundance, UT, 1989) Proc. Sympos. Pure Math., vol. 53, Amer. Math. Soc., Providence, RI, 1991, pp. 175–221. MR 1141202, DOI 10.1090/pspum/053/1141202
Richard M. Hain, Completions of mapping class groups and the cycle $C-C^-$, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 75–105. MR 1234261, DOI 10.1090/conm/150/01287
Richard Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), no. 3, 597–651. MR 1431828, DOI 10.1090/S0894-0347-97-00235-X
Wilberd van der Kallen, Homology stability for linear groups, Invent. Math. 60 (1980), no. 3, 269–295. MR 586429, DOI 10.1007/BF01390018
A. Knapp, Lie groups, Lie algebras, and cohomology, Princeton University Press, Princeton, NJ, 1988. MR:89j:22034
K. Knudson, The homology of special linear groups over polynomial rings, Ann. Sci. Ecole Norm. Sup. (4) 30 (1997), 385–416.
Alexander Lubotzky and Andy R. Magid, Cohomology, Poincaré series, and group algebras of unipotent groups, Amer. J. Math. 107 (1985), no. 3, 531–553. MR 789654, DOI 10.2307/2374368
Daniel G. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411–418. MR 231919, DOI 10.1016/0021-8693(68)90069-0
Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
M. S. Raghunathan, Cohomology of arithmetic subgroups of algebraic groups. I, II, Ann. of Math. (2) 86 (1967), 409–424; 87 (1967), 279–304. MR 0227313, DOI 10.2307/1970585
A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477 (Russian). MR 0472792