Finitely presentable, non-Hopfian groups with Kazhdan’s Property (T) and infinite outer automorphism group
HTML articles powered by AMS MathViewer
- by Yves de Cornulier PDF
- Proc. Amer. Math. Soc. 135 (2007), 951-959 Request permission
Erratum: Proc. Amer. Math. Soc. 139 (2011), 383-384.
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 $p$-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answers a question of Ollivier and Wise.References
- Herbert Abels, An example of a finitely presented solvable group, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 205–211. MR 564423
- Herbert Abels, Finite presentability of $S$-arithmetic groups. Compact presentability of solvable groups, Lecture Notes in Mathematics, vol. 1261, Springer-Verlag, Berlin, 1987. MR 903449, DOI 10.1007/BFb0079708
- Helmut Behr, $\textrm {SL}_{3}(\textbf {F}_{q}[t])$ is not finitely presentable, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 213–224. MR 564424
- Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
- I. Belegradek, A. Szczepański. Endomorphisms of relatively hyperbolic groups. Preprint 2005, arXiv math.GR/0501321.
- P. Hall, The Frattini subgroups of finitely generated groups, Proc. London Math. Soc. (3) 11 (1961), 327–352. MR 124406, DOI 10.1112/plms/s3-11.1.327
- Pierre de la Harpe and Alain Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175 (1989), 158 (French, with English summary). With an appendix by M. Burger. MR 1023471
- Martin Kneser, Erzeugende und Relationen verallgemeinerter Einheitengruppen, J. Reine Angew. Math. 214(215) (1964), 345–349 (German). MR 161863, DOI 10.1515/crll.1964.214-215.345
- G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
- Y. Ollivier, D. Wise. Kazhdan groups with infinite outer automorphism group. Preprint 2005, arXiv math.GR/0409203; to appear in Trans. Amer. Math. Soc.
- Frédéric Paulin, Outer automorphisms of hyperbolic groups and small actions on $\textbf {R}$-trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 331–343. MR 1105339, DOI 10.1007/978-1-4612-3142-4_{1}2
- U. Rehmann and C. Soulé, Finitely presented groups of matrices, Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976, pp. 164–169. MR 0486175
- Yehuda Shalom, Bounded generation and Kazhdan’s property (T), Inst. Hautes Études Sci. Publ. Math. 90 (1999), 145–168 (2001). MR 1813225
- S. P. Wang, On the Mautner phenomenon and groups with property $(\textrm {T})$, Amer. J. Math. 104 (1982), no. 6, 1191–1210. MR 681733, DOI 10.2307/2374057
- 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.
- Andrzej Żuk, La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 5, 453–458 (French, with English and French summaries). MR 1408975
Additional Information
- Yves de Cornulier
- Affiliation: Institut de Géométrie, Algèbre et Topologie (IGAT), École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
- MR Author ID: 766953
- Email: decornul@clipper.ens.fr
- Received by editor(s): February 25, 2005
- Received by editor(s) in revised form: October 28, 2005
- Published electronically: September 26, 2006
- Communicated by: Dan M. Barbasch
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 951-959
- MSC (2000): Primary 20F28; Secondary 20G25, 17B56
- DOI: https://doi.org/10.1090/S0002-9939-06-08588-1
- MathSciNet review: 2262894