Relative completions of linear groups

over and

Author:
Kevin P. Knudson

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

MSC (1991):
Primary 55P60, 20G35, 20H05; Secondary 20G10, 20F14

Published electronically:
July 26, 1999

MathSciNet review:
1641103

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Abstract | References | Similar Articles | Additional Information

Abstract: We compute the completion of the groups and

relative to the obvious homomorphisms to ; this is a generalization of the classical Malcev completion. We also make partial computations of the rational second cohomology of these groups.

**1.**Armand Borel,*Topology of Lie groups and characteristic classes*, Bull. Amer. Math. Soc.**61**(1955), 397–432. MR**0072426**, 10.1090/S0002-9904-1955-09936-1**2.**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****3.**A. K. Bousfield,*Homological localization towers for groups and Π-modules*, Mem. Amer. Math. Soc.**10**(1977), no. 186, vii+68. MR**0447375****4.**Fritz Grunewald, Jens Mennicke, and Leonid Vaserstein,*On the groups 𝑆𝐿₂(𝑍[𝑥]) and 𝑆𝐿₂(𝑘[𝑥,𝑦])*, Israel J. Math.**86**(1994), no. 1-3, 157–193. MR**1276133**, 10.1007/BF02773676**5.**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**, 10.1090/pspum/053/1141202**6.**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**, 10.1090/conm/150/01287**7.**Richard Hain,*Infinitesimal presentations of the Torelli groups*, J. Amer. Math. Soc.**10**(1997), no. 3, 597–651. MR**1431828**, 10.1090/S0894-0347-97-00235-X**8.**Wilberd van der Kallen,*Homology stability for linear groups*, Invent. Math.**60**(1980), no. 3, 269–295. MR**586429**, 10.1007/BF01390018**9.**A. Knapp,*Lie groups, Lie algebras, and cohomology*, Princeton University Press, Princeton, NJ, 1988. MR:89j:22034**10.**K. Knudson,*The homology of special linear groups over polynomial rings*, Ann. Sci. Ecole Norm. Sup. (4)**30**(1997), 385-416.**11.**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**, 10.2307/2374368**12.**Daniel G. Quillen,*On the associated graded ring of a group ring*, J. Algebra**10**(1968), 411–418. MR**0231919****13.**Daniel Quillen,*Rational homotopy theory*, Ann. of Math. (2)**90**(1969), 205–295. MR**0258031****14.**M. S. Raghunathan,*Cohomology of arithmetic subgroups of algebraic groups. I, II*, Ann. of Math. (2) 86 (1967), 409-424; ibid. (2)**87**(1967), 279–304. MR**0227313****15.**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**

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Additional Information

**Kevin P. Knudson**

Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208

Address at time of publication:
Department of Mathematics, Wayne State University, Detroit, Michigan 48202

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

Received by editor(s):
January 20, 1998

Published electronically:
July 26, 1999

Additional Notes:
Supported by an NSF Postdoctoral Fellowship, grant no. DMS-9627503

Article copyright:
© Copyright 2000
American Mathematical Society