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