## Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian

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- by Andrew Sale and Tim Susse PDF
- Trans. Amer. Math. Soc.
**372**(2019), 7785-7803 Request permission

## Abstract:

We generalize the notion of a separating intersection of links (SIL) to give necessary and sufficient criteria on the defining graph $\Gamma$ of a right-angled Coxeter group $W_\Gamma$ so that its outer automorphism group is large: that is, it contains a finite index subgroup that admits the free group $F_2$ as a quotient. When $\operatorname {Out}(W_\Gamma )$ is not large, we show it is virtually abelian. We also show that the same dichotomy holds for the outer automorphism groups of graph products of finite abelian groups. As a consequence, these groups have property (T) if and only if they are finite or equivalently $\Gamma$ contains no SIL.## References

- Javier Aramayona and Conchita Martínez-Pérez,
*On the first cohomology of automorphism groups of graph groups*, J. Algebra**452**(2016), 17–41. MR**3461054**, DOI 10.1016/j.jalgebra.2015.11.045 - Bachir Bekka, Pierre de la Harpe, and Alain Valette,
*Kazhdan’s property (T)*, New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR**2415834**, DOI 10.1017/CBO9780511542749 - Jason Behrstock, Mark F. Hagen, and Alessandro Sisto,
*Thickness, relative hyperbolicity, and randomness in Coxeter groups*, Algebr. Geom. Topol.**17**(2017), no. 2, 705–740. With an appendix written jointly with Pierre-Emmanuel Caprace. MR**3623669**, DOI 10.2140/agt.2017.17.705 - L. J. Corredor and M. A. Gutierrez,
*A generating set for the automorphism group of a graph product of abelian groups*, Internat. J. Algebra Comput.**22**(2012), no. 1, 1250003, 21. MR**2900856**, DOI 10.1142/S0218196711006698 - Donald J. Collins,
*The automorphism group of a free product of finite groups*, Arch. Math. (Basel)**50**(1988), no. 5, 385–390. MR**942533**, DOI 10.1007/BF01196497 - Ruth Charney, Kim Ruane, Nathaniel Stambaugh, and Anna Vijayan,
*The automorphism group of a graph product with no SIL*, Illinois J. Math.**54**(2010), no. 1, 249–262. MR**2776995** - Michael W. Davis,
*The geometry and topology of Coxeter groups*, Introduction to modern mathematics, Adv. Lect. Math. (ALM), vol. 33, Int. Press, Somerville, MA, 2015, pp. 129–142. MR**3445448** - Matthew B. Day,
*On solvable subgroups of automorphism groups of right-angled Artin groups*, Internat. J. Algebra Comput.**21**(2011), no. 1-2, 61–70. MR**2787453**, DOI 10.1142/S021819671100608X - N. D. Gilbert,
*Presentations of the automorphism group of a free product*, Proc. London Math. Soc. (3)**54**(1987), no. 1, 115–140. MR**872253**, DOI 10.1112/plms/s3-54.1.115 - Fritz Grunewald and Alexander Lubotzky,
*Linear representations of the automorphism group of a free group*, Geom. Funct. Anal.**18**(2009), no. 5, 1564–1608. MR**2481737**, DOI 10.1007/s00039-009-0702-2 - Fritz Grunewald, Michael Larsen, Alexander Lubotzky, and Justin Malestein,
*Arithmetic quotients of the mapping class group*, Geom. Funct. Anal.**25**(2015), no. 5, 1493–1542. MR**3426060**, DOI 10.1007/s00039-015-0352-5 - Mauricio Gutierrez, Adam Piggott, and Kim Ruane,
*On the automorphisms of a graph product of abelian groups*, Groups Geom. Dyn.**6**(2012), no. 1, 125–153. MR**2888948**, DOI 10.4171/GGD/153 - Vincent Guirardel and Andrew Sale,
*Vastness properties of automorphism groups of RAAGs*, J. Topol.**11**(2018), no. 1, 30–64. MR**3784226**, DOI 10.1112/topo.12047 - Marek Kaluba, Piotr W. Nowak, and Narutaka Ozawa,
*$\mathrm {Aut}(\mathbb {F}_5)$ has property $(T)$*, Mathematische Annalen , posted on (Aug. 8 2019)., DOI 10.1007/s00208-019-01874-9 - Marek Kaluba, Dawid Kielak, and Piotr W. Nowak,
*On property $(T)$ for $\mathrm {Aut} (F_n)$ and $\mathrm {SL}_n (\mathbb {Z})$*, preprint, arXiv:1812.03456, 2018. - Michael R. Laurence.
*Automorphisms of graph products of groups*. PhD thesis, QMW College, University of London, 1992. - Michael R. Laurence,
*A generating set for the automorphism group of a graph group*, J. London Math. Soc. (2)**52**(1995), no. 2, 318–334. MR**1356145**, DOI 10.1112/jlms/52.2.318 - Gabor Moussong,
*Hyperbolic Coxeter groups*, ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.)–The Ohio State University. MR**2636665** - Bernhard Mühlherr,
*Automorphisms of graph-universal Coxeter groups*, J. Algebra**200**(1998), no. 2, 629–649. MR**1610676**, DOI 10.1006/jabr.1997.7230 - Herman Servatius,
*Automorphisms of graph groups*, J. Algebra**126**(1989), no. 1, 34–60. MR**1023285**, DOI 10.1016/0021-8693(89)90319-0 - Jacques Tits,
*Sur le groupe des automorphismes de certains groupes de Coxeter*, J. Algebra**113**(1988), no. 2, 346–357 (French). MR**929765**, DOI 10.1016/0021-8693(88)90164-0

## Additional Information

**Andrew Sale**- Affiliation: Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall (Keller Hall 401A), Honolulu, Hawaii 96822
- MR Author ID: 1075109
**Tim Susse**- Affiliation: Department of Mathematics, Bard College at Simon’s Rock, Great Barrington, Massachusetts 01230
- MR Author ID: 1126468
- Received by editor(s): September 19, 2017
- Received by editor(s) in revised form: February 8, 2019
- Published electronically: September 10, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 7785-7803 - MSC (2010): Primary 20E36, 20F28, 20F55, 20F65
- DOI: https://doi.org/10.1090/tran/7897
- MathSciNet review: 4029681