Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: August 4, 2004
MathSciNet review: 2093062
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the groups ${\mathcal{G}}$ whose integral group rings have hyperbolic unit groups ${\mathcal{U}(\mathbb{Z} {\mathcal{G}}) }$. We classify completely the torsion subgroups of $\mathcal{U}(\mathbb{Z} {\mathcal{G}})$ and the polycyclic-by-finite subgroups of the group ${\mathcal{G}}$. Finally, we classify the groups for which the boundary of ${\mathcal{U}(\mathbb{Z} {\mathcal{G}}) }$ has dimension zero.

References [Enhancements On Off] (What's this?)

  • 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 $1+\Delta (G)\Delta (A)$ is torsion free, J. Group Theory 6 (2003), 223-228.MR 2004a:16043
  • 10. Ol'Shanski{\u{\i}}\kern.15em, 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20C07, 16S34, 20F67

Retrieve articles in all journals with MSC (2000): 20C07, 16S34, 20F67

Additional Information

S. O. Juriaans
Affiliation: Instituto de Matemática e Estatística, CP. 666.281, CEP.05315-970, São Paulo, Brazil

I. B. S. Passi
Affiliation: Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India

Dipendra Prasad
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India

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
Published electronically: 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