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Hyperbolic unit groups
Authors:
S. O. Juriaans, I. B. S. Passi and Dipendra Prasad
Journal:
Proc. Amer. Math. Soc. 133 (2005), 415-423
MSC (2000):
Primary 20C07, 16S34, 20F67
Posted:
August 4, 2004
MathSciNet review:
2093062
Full-text PDF Free Access
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Abstract: In this paper we study the groups  whose integral group rings have hyperbolic unit groups  . We classify completely the torsion subgroups of  and the polycyclic-by-finite subgroups of the group  . Finally, we classify the groups for which the boundary of  has dimension zero.
- 1.
M.
Coornaert, T.
Delzant, and A.
Papadopoulos, Géométrie et théorie des
groupes, Lecture Notes in Mathematics, vol. 1441,
Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de
Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994
(92f:57003)
- 2.
I.
Martin Isaacs, Character theory of finite groups, Academic
Press [Harcourt Brace Jovanovich Publishers], New York, 1976. Pure and
Applied Mathematics, No. 69. MR 0460423
(57 #417)
- 3.
É.
Ghys and P.
de la Harpe (eds.), Sur les groupes hyperboliques
d’après Mikhael Gromov, Progress in Mathematics,
vol. 83, Birkhäuser Boston Inc., Boston, MA, 1990 (French).
Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648
(92f:53050)
- 4.
M.
Gromov, Hyperbolic groups, Essays in group theory, Math. Sci.
Res. Inst. Publ., vol. 8, Springer, New York, 1987,
pp. 75–263. MR 919829
(89e:20070), http://dx.doi.org/10.1007/978-1-4613-9586-7_3
- 5.
B.
Huppert, Endliche Gruppen. I, Die Grundlehren der
Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin, 1967
(German). MR
0224703 (37 #302)
- 6.
Eric
Jespers, Free normal complements and the unit
group of integral group rings, Proc. Amer.
Math. Soc. 122 (1994), no. 1, 59–66. MR 1221725
(94k:16058), http://dx.doi.org/10.1090/S0002-9939-1994-1221725-1
- 7.
Eric
Jespers, Units in integral group rings: a survey, Methods in
ring theory (Levico Terme, 1997) Lecture Notes in Pure and Appl. Math.,
vol. 198, Dekker, New York, 1998, pp. 141–169. MR 1767976
(2001d:16048)
- 8.
Ernst
Kleinert, Units of classical orders: a survey, Enseign. Math.
(2) 40 (1994), no. 3-4, 205–248. MR 1309127
(95k:11151)
- 9.
Zbigniew
Marciniak and Sudarshan
K. Sehgal, The unit group of 1+Δ(𝐺)Δ(𝐴)
is torsion-free, J. Group Theory 6 (2003),
no. 2, 223–228. MR 1961569
(2004a:16043), http://dx.doi.org/10.1515/jgth.2003.016
- 10.
A.
Yu. Ol′shanskiĭ, Almost every group is
hyperbolic, Internat. J. Algebra Comput. 2 (1992),
no. 1, 1–17. MR 1167524
(93j:20068), http://dx.doi.org/10.1142/S0218196792000025
- 11.
Inder
Bir S. Passi, Group rings and their augmentation ideals,
Lecture Notes in Mathematics, vol. 715, Springer, Berlin, 1979. MR 537126
(80k:20009)
- 12.
Donald
S. Passman, The algebraic structure of group rings, Pure and
Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York,
1977. MR
470211 (81d:16001)
- 13.
Sudarshan
K. Sehgal, Topics in group rings, Monographs and Textbooks in
Pure and Applied Math., vol. 50, Marcel Dekker Inc., New York, 1978.
MR 508515
(80j:16001)
- 14.
S.
K. Sehgal, Units in integral group rings, Pitman Monographs
and Surveys in Pure and Applied Mathematics, vol. 69, Longman
Scientific & Technical, Harlow, 1993. With an appendix by Al Weiss. MR 1242557
(94m:16039)
- 15.
M.
Shirvani and B.
A. F. Wehrfritz, Skew linear groups, London Mathematical
Society Lecture Note Series, vol. 118, Cambridge University Press,
Cambridge, 1986. MR 883801
(89h:20001)
- 1.
- Coornaert, M., Delzant, T., Papadopoulos, A. : Géométrie et théorie des groupes, LNM 1441, Springer-Verlag, 1990. MR 92f:57003
- 2.
- Isaac, I.M. : Character Theory of Finite Groups, Academic Press, New York, 1976. MR 57:417
- 3.
- Ghys, E. and Harpe, P. de la (Editors) : Sur les Groupes Hyperboliques d'après Mikhael Gromov, Progr. Math. Vol. 83, Birkhäuser, 1990. MR 92f:53050
- 4.
- Gromov, M. : Hyperbolic groups, In: Essays in group theory (S. M. Gersten, Ed.), Springer Verlag, MSRI Publ. 8, 1997, 75-263.MR 89e:20070
- 5.
- Huppert, B. : Endliche Gruppen I, Springer-Verlag, 1967. MR 37:302
- 6.
- Jespers, E. : Free normal complements and the unit group of integral group rings, Proc. AMS vol. 122 (1994), 59-66. MR 94k:16058
- 7.
- Jespers, E. : Units in integral group rings: a survey, Methods in Ring Theory, Lecture Notes in Pure and Applied Mathematics, 198, Dekker, New York, (1998), 141-169. MR 2001d:16048
- 8.
- Kleinert, E. : Units of classical orders: a survey, L'Enseignement Math. 40 (1994), 205-248. MR 95k:11151
- 9.
- Marciniak, Z., and Sehgal S.K.: The unit group of
 is torsion free, J. Group Theory 6 (2003), 223-228.MR 2004a:16043
- 10.
- Ol'Shanski
 , A.Yu. : Almost every group is hyperbolic, Internat. J. Algebra Comput. 2 (1992), 1-17. MR 93j:20068
- 11.
- Passi, I.B.S. : Group Rings and Their Augmentation Ideals, Lecture Notes in Mathematics 715, Springer, New York, 1979. MR 80k:20009
- 12.
- Passman, D.S. : Algebraic Structure of Group Rings, Interscience, New York, 1977. MR 81d:16001
- 13.
- Sehgal, S.K. : Topics in Group Rings, Marcel Dekker, New York and Basel, 1978. MR 80j:16001
- 14.
- Sehgal, S.K. : Units in Integral Group Rings, Longman, Essex, 1993. MR 94m:16039
- 15.
- Shirvani, M., Wehrfritz, B.A.F. : Skew Linear Groups, London Math. Soc. Lecture Notes Series 118, Cambridge, 1986. MR 89h:20001
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Additional Information
S. O. Juriaans
Affiliation:
Instituto de Matemática e Estatística, CP. 666.281, CEP.05315-970, São Paulo, Brazil
Email:
ostanley@ime.usp.br
I. B. S. Passi
Affiliation:
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India
Email:
passi@mri.ernet.in
Dipendra Prasad
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India
Email:
dprasad@math.tifr.res.in
DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07578-1
PII:
S 0002-9939(04)07578-1
Keywords:
Hyperbolic group,
group ring,
unit group,
Wedderburn decomposition.
Received by editor(s):
March 20, 2003
Received by editor(s) in revised form:
October 18, 2003
Posted:
August 4, 2004
Additional Notes:
This research was partially supported by CNPq-Brazil, FAPESP-Brazil.
Communicated by:
Jonathan I. Hall
Article copyright:
© Copyright 2004 American Mathematical Society
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