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Transactions of the American Mathematical Society

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Kazhdan groups with infinite outer automorphism group


Authors: Yann Ollivier and Daniel T. Wise
Journal: Trans. Amer. Math. Soc. 359 (2007), 1959-1976
MSC (2000): Primary 20F28, 20F06, 20E22, 20P05
DOI: https://doi.org/10.1090/S0002-9947-06-03941-9
Published electronically: November 17, 2006
MathSciNet review: 2276608
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Abstract | References | Similar Articles | Additional Information

Abstract: For each countable group $ Q$ we produce a short exact sequence $ 1\rightarrow N \rightarrow G \rightarrow Q\rightarrow 1$ where $ G$ has a graphical $ \frac16$ presentation and $ N$ is f.g. and satisfies property $ T$.

As a consequence we produce a group $ N$ with property $ T$ such that $ \operatorname{Out}(N)$ is infinite.

Using the tools developed we are also able to produce examples of non-Hopfian and non-coHopfian groups with property $ T$.

One of our main tools is the use of random groups to achieve certain properties.


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

  • [BW] Inna Bumagin and Daniel T. Wise, Every group is an outer automorphism group of a finitely generated group, J. Pure Appl. Algebra 200 (2005), no. 1-2, 137-147. MR 2142354 (2005m:20085)
  • [Cor] Yves de Cornulier, Finitely presentable, non-Hopfian groups with Kazhdan's property (T) and infinite outer automorphism group, To appear in Proc. Amer. Math. Soc.
  • [DSV03] Giuliana Davidoff, Peter Sarnak, and Alain Valette, Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, vol. 55, Cambridge University Press, Cambridge, 2003. MR 1989434 (2004f:11001)
  • [dlHV89] Pierre de la Harpe and Alain Valette, La propriété $ (T)$ de Kazhdan pour les groupes localement compacts, Astérisque, no. 175, Soc. Math. Fr., 1989, with an appendix by M. Burger. MR 1023471 (90m:22001)
  • [GdlH90] 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, papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648 (92f:53050)
  • [Ghy03] Étienne Ghys, Groupes aléatoires (d'après Misha Gromov,$ \dots$), Astérisque (2004), no. 294, 173-204, Séminaire Bourbaki, Vol. 2003/2004, Exp. 916. MR 2111644 (2005j:20049)
  • [Gro87] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75-263. MR 0919829 (89e:20070)
  • [Gro93] -, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991), Cambridge Univ. Press, Cambridge, 1993, pp. 1-295. MR 1253544 (95m:20041)
  • [Gro03] -, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73-146. MR 1978492 (2004j:20088a)
  • [HW04] Frédéric Haglund and Daniel T. Wise, Special cube complexes, preprint, 2004.
  • [Kes59] Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336-354. MR 0109367 (22:253)
  • [Lub94] Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Birkhäuser Verlag, Basel, 1994, with an appendix by Jonathan D. Rogawski. MR 1308046 (96g:22018)
  • [LS77] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin, 1977, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064 (58:28182)
  • [NR97] Graham Niblo and Lawrence Reeves, Groups acting on $ {{\rm CAT}(0)}$ cube complexes, Geom. Topol. 1 (1997), approx. 7 pp. (electronic). MR 1432323 (98d:57005)
  • [NR98] Graham A. Niblo and Martin A. Roller, Groups acting on cubes and Kazhdan's property (T), Proc. Amer. Math. Soc. 126 (1998), no. 3, 693-699. MR 1459140 (98k:20058)
  • [Oll03a] Y. Ollivier, On a small cancellation theorem of Gromov, to appear in Bull. Belgian Math. Soc.
  • [Oll03b] -, Cayley graphs containing expanders, after Gromov, expository manuscript, 2003.
  • [Oll04] -, Sharp phase transition theorems for hyperbolicity of random groups, GAFA, Geom. Funct. Anal. 14 (2004), no. 3, 595-679. MR 2100673 (2005m:20101)
  • [Oll05] -, A January 2005 invitation to random groups, Ensaios Matemáticos 10, Sociedade Brasileira de Matemática, Rio de Janeiro, 2005. MR 2205306
  • [Pau91] Frédéric Paulin, Outer automorphisms of hyperbolic groups and small actions on $ {\bf R}$-trees, Arboreal group theory (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 331-343. MR 1105339 (92g:57003)
  • [Rip82] E. Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), no. 1, 45-47. MR 0642423 (83c:20049)
  • [Sil03] L. Silberman, Addendum to ``Random walk in random groups'' by M. Gromov, GAFA, Geom. Funct. Anal. 13 (2003), no. 1, 147-177. MR 1978493 (2004j:20088b)
  • [Val04] Alain Valette, Nouvelles approches de la propriété (T) de Kazhdan, Astérisque (2004), no. 294, 97-124, Séminaire Bourbaki, Vol. 2003/2004, Exp. 913. MR 2111641 (2005j:22003)
  • [Wis98] Daniel T. Wise, Incoherent negatively curved groups, Proc. Amer. Math. Soc. 126 (1998), no. 4, 957-964. MR 1423338 (98f:20016)
  • [Wis03] -, A residually finite version of Rips's construction, Bull. London Math. Soc. 35 (2003), no. 1, 23-29. MR 1934427 (2003g:20047)
  • [Wis04] Daniel T. Wise, Cubulating small cancellation groups, GAFA, Geom. Funct. Anal. 14 (2004), no. 1, 150-214. MR 2053602 (2005c:20069)
  • [Zuk03] A. Zuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), no. 3, 643-670. MR 1995802 (2004m:20079)

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Additional Information

Yann Ollivier
Affiliation: CNRS, UMPA, École normale supérieure de Lyon, 46, allée d’Italie, 69364 Lyon cedex 7, France
Email: yann.ollivier@umpa.ens-lyon.fr

Daniel T. Wise
Affiliation: Department of Mathematics, McGill University, Montréal, Québec, Canada H3A 2K6
Email: wise@math.mcgill.ca

DOI: https://doi.org/10.1090/S0002-9947-06-03941-9
Keywords: Outer automorphism groups, property $T$, small cancellation, random groups
Received by editor(s): September 27, 2004
Received by editor(s) in revised form: January 10, 2005
Published electronically: November 17, 2006
Additional Notes: This research was partially supported by NSERC grant
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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