Kazhdan groups with infinite outer automorphism group

Ollivier, Yann; Wise, Daniel

doi:10.1090/S0002-9947-06-03941-9

Kazhdan groups with infinite outer automorphism group
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by Yann Ollivier and Daniel T. Wise PDF

Trans. Amer. Math. Soc. 359 (2007), 1959-1976 Request permission

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 $\frac 16$ 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

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, DOI 10.1016/j.jpaa.2004.12.033
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.
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, DOI 10.1017/CBO9780511615825
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
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, DOI 10.1007/978-1-4684-9167-8
Étienne Ghys, Groupes aléatoires (d’après Misha Gromov,$\dots$), Astérisque 294 (2004), viii, 173–204 (French, with French summary). MR 2111644
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73–146. MR 1978492, DOI 10.1007/s000390300002
Frédéric Haglund and Daniel T. Wise, Special cube complexes, preprint, 2004.
Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. MR 109367, DOI 10.1090/S0002-9947-1959-0109367-6
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, DOI 10.1007/978-3-0346-0332-4
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Graham Niblo and Lawrence Reeves, Groups acting on $\textrm {CAT}(0)$ cube complexes, Geom. Topol. 1 (1997), approx. 7 pp.}, issn=1465-3060, review= MR 1432323, doi=10.2140/gt.1997.1.1,
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, DOI 10.1090/S0002-9939-98-04463-3
Y. Ollivier, On a small cancellation theorem of Gromov, to appear in Bull. Belgian Math. Soc.
—, Cayley graphs containing expanders, after Gromov, expository manuscript, 2003.
Y. Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, Geom. Funct. Anal. 14 (2004), no. 3, 595–679. MR 2100673, DOI 10.1007/s00039-004-0470-y
Yann Ollivier, A January 2005 invitation to random groups, Ensaios Matemáticos [Mathematical Surveys], vol. 10, Sociedade Brasileira de Matemática, Rio de Janeiro, 2005. MR 2205306
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
E. Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), no. 1, 45–47. MR 642423, DOI 10.1112/blms/14.1.45
M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73–146. MR 1978492, DOI 10.1007/s000390300002
Alain Valette, Nouvelles approches de la propriété (T) de Kazhdan, Astérisque 294 (2004), vii, 97–124 (French, with French summary). MR 2111641
Daniel T. Wise, Incoherent negatively curved groups, Proc. Amer. Math. Soc. 126 (1998), no. 4, 957–964. MR 1423338, DOI 10.1090/S0002-9939-98-04146-X
Daniel T. Wise, A residually finite version of Rips’s construction, Bull. London Math. Soc. 35 (2003), no. 1, 23–29. MR 1934427, DOI 10.1112/S0024609302001406
D. T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1, 150–214. MR 2053602, DOI 10.1007/s00039-004-0454-y
A. Żuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), no. 3, 643–670. MR 1995802, DOI 10.1007/s00039-003-0425-8