Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Finitely presentable, non-Hopfian groups with Kazhdan's Property (T) and infinite outer automorphism group

Author(s): Yves de Cornulier
Journal: Proc. Amer. Math. Soc. 135 (2007), 951-959.
MSC (2000): Primary 20F28; Secondary 20G25, 17B56
Posted: September 26, 2006
Errata: Proc. Amer. Math. Soc. 139 (2011), 383-384
MathSciNet review: 2262894
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We give simple examples of Kazhdan groups with infinite outer automorphism groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. As arithmetic lattices in (non-semisimple) Lie groups, our examples are in addition finitely presented.

We also use results of Abels about compact presentability of -adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answers a question of Ollivier and Wise.

References:

[A1]

H. ABELS. An example of a finitely presentable solvable group. In: ``Homological group theory,'' Proceedings Symposium, Durham, 1977, London Math. Soc. Lecture Note Ser. 36 (1979), 105-211. MR 0564423 (82b:20047)

[A2]

H. ABELS. Finite presentability of -arithmetic groups. Compact presentability of solvable groups. Lecture Notes in Math. 1261, Springer, 1987. MR 0903449 (89b:22017)

[Be]

H. BEHR. Chevalley groups of rank 2 over $\mathbf{F}_q(t)$ are not finitely presentable. In: ``Homological group theory,'' Proceedings Symposium, Durham, 1977, London Math. Soc. Lecture Note Ser. 36 (1979), 213-224. MR 0564424 (81e:20052)

[BHC]

A. BOREL, HARISH-CHANDRA. Arithmetic subgroups of algebraic groups. Ann. of Math. 75 (1962), 485-535. MR 0147566 (26:5081)

[BS]

I. BELEGRADEK, A. SZCZEPANSKI. Endomorphisms of relatively hyperbolic groups. Preprint 2005, arXiv math.GR/0501321.

[Ha]

P. HALL. The Frattini subgroups of finitely generated groups. Proc. London Math. Soc. (3) 11 (1961), 327-352. MR 0124406 (23:A1718)

[HV]

P. DE LA HARPE, Alain VALETTE. La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175, SMF, 1989. MR 1023471 (90m:22001)

[Kn]

M. KNESER. Erzeugende und Relationen verallgemeinerter Einheitengruppen. J. Reine Angew. Math. 214/215 (1964), 345-349. MR 0161863 (28:5067)

[Ma]

G. MARGULIS. Discrete Subgroups of Semisimple Lie Groups. Springer, 1991. MR 1090825 (92h:22021)

[OW]

Y. OLLIVIER, D. WISE. Kazhdan groups with infinite outer automorphism group. Preprint 2005, arXiv math.GR/0409203; to appear in Trans. Amer. Math. Soc.

[Pa]

F. PAULIN. Outer automorphisms of hyperbolic groups and small actions on R-trees. In ``Arboreal Group Theory'' (MSRI, Berkeley, September, 1988) R.C. Alperin ed., M.S.R.I., Pub. 19, Springer-Verlag, 1991. MR 1105339 (92g:57003)

[RS]

U. REHRMANN, C. SOULÉ. Finitely presented groups of matrices. In ``Algebraic K-theory" Proceedings Conference, Evanston, 1976. Lecture Notes in Math. 551: 164-169, Springer, 1976. MR 0486175 (58:5955)

[Sh]

Y. SHALOM. Bounded generation and Kazhdan's property (T). Publ. Math. Inst. Hautes Études Sci. 90 (1999), 145-168. MR 1813225 (2001m:22030)

[Wa]

S. P. WANG. On the Mautner phenomenon and groups with property (T). Amer. J. Math. 104(6) (1982),

1191-1210. MR 0681733 (84g:22033)

[Wo]

Report of the workshop Geometrization of Kazhdan's Property (T) (organizers: B. Bekka, P. de la Harpe, A. Valette; 2001). Unpublished; currently available at http://www.mfo. de/cgi-bin/tagungsdb?type=21&tnr=0128a.

[Zu]

A. ZUK. La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres. C. R. Math. Acad. Sci. Paris 323, Série I (1996), 453-458. MR 1408975 (97i:22001)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|