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CONJUGATING AUTOMORPHISMS OF GRAPH PRODUCTS: KAZHDAN’S PROPERTY (T) AND SQ-UNIVERSALITY

Part of: Graph theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  07 August 2019

ANTHONY GENEVOIS Open the ORCID record for ANTHONY GENEVOIS [Opens in a new window]  and
OLGA VARGHESE Open the ORCID record for OLGA VARGHESE [Opens in a new window]
ANTHONY GENEVOIS*
Affiliation:
Département de Mathématiques, Bâtiment 307, Faculté des Sciences d’Orsay, Université Paris-Sud, F-91405 Orsay, France email anthony.genevois@math.u-psud.fr
OLGA VARGHESE
Affiliation:
Department of Mathematics, Münster University, Einsteinstraße 62, 48149 Münster, Germany email olga.varghese@uni-muenster.de
*
✉anthony.genevois@math.u-psud.fr
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Abstract

An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan’s property (T) and when it satisfies some vastness properties including SQ-universality.

Keywords

graph product of groupsconjugating automorphismKazhdan’s property (T)SQ-universality

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20F28: Automorphism groups of groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 2 , April 2020 , pp. 272 - 282
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by a public grant as part of the Fondation Mathématique Jacques Hadamard. The second author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044, Mathematics Münster: Dynamics–Geometry–Structure.

References

Aramayona, J. and Martinez-Perez, C., ‘On the first cohomology of automorphism groups of graph groups’, J. Algebra 452 (2016), 17–41.Google Scholar
Bekka, B., de la Harpe, P. and Valette, A., Kazhdan’s Property (T), New Mathematical Monographs, 11 (Cambridge University Press, Cambridge, 2008).Google Scholar
Bestvina, M. and Fujiwara, K., ‘Quasi-homomorphisms on mapping class groups’, Glas. Mat. Ser. III 42(62) (2017), 213–236.Google Scholar
Dahmani, F., Guirardel, V. and Osin, D., Hyperbolically Embedded Subgroups and Rotating Families in Groups Acting on Hyperbolic Spaces, Memoirs of the American Mathematical Society, 245(1156) (American Mathematical Society, Providence, RI, 2017).Google Scholar
Damiani, C., ‘A journey through loop braid groups’, Expo. Math. 35(3) (2017), 252–285.Google Scholar
Davis, M., The Geometry and Topology of Coxeter Groups, London Mathematical Society Monographs, 32 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
Fouxe-Rabinovich, D., ‘Über die Automorphismengruppen der freien Produkte I’, Mat. Sb. 50(2) (1940), 265–276.Google Scholar
Fouxe-Rabinovich, D., ‘Über die Automorphismengruppen der freien Produkte II’, Mat. Sb. 51(1) (1941), 183–220.Google Scholar
Genevois, A. and Martin, A., ‘Automorphisms of graph products of groups from a geometric perspective’, Proc. Lond. Math. Soc., to appear.Google Scholar
Goldsmith, D., ‘The theory of motion groups’, Michigan Math. J. 28(1) (1981), 3–17.Google Scholar
Guirardel, V. and Sale, A., ‘Vastness properties of automorphism groups of RAAGs’, J. Topol. 11(1) (2018), 30–64.Google Scholar
Gutierrez, M., Piggot, A. and Ruane, K., ‘On the automorphisms of a graph product of abelian groups’, Groups Geom. Dyn. 6 (1980), 125–153.Google Scholar
Kaluba, M., Kielak, D. and Nowak, P., ‘On property (T) for $\text{Aut}(F_{n})$ and $\text{SL}_{n}(\mathbb{Z})$’, Preprint, 2018, arXiv:1812.03456.Google Scholar
Kaluba, M., Nowak, P. and Ozawa, N., ‘$\text{Aut}(F_{5})$ has property (T)’, Preprint, 2017,arXiv:1712.07167.Google Scholar
Karrass, P., Pietrowski, A. and Solitar, D., ‘Automorphisms of a free product with an amalgamated subgroup’, Contrib. Group Theory 33 (1980), 328–340.Google Scholar
Laurence, M., ‘A generating set for the automorphism group of a graph group’, J. Lond. Math. Soc. (2) 52(2) (1995), 318–334.Google Scholar
Leder, N., ‘Serre’s Property FA for automorphism groups of free products’, Preprint, 2018.Google Scholar
Leder, N. and Varghese, O., ‘On property (T) for automorphism groups of graph products’, Preprint, 2019, arxiv1902.04370.Google Scholar
McCullough, D. and Miller, A., Symmetric Automorphisms of Free Products, Memoirs of the American Mathematical Society, 582 (American Mathematical Society, Providence, RI, 1996).Google Scholar
Minasyan, A. and Osin, D., ‘Acylindrical hyperbolicity of groups acting on trees’, Math. Ann. 362(3–4) (2015), 1055–1105.Google Scholar
Neumann, P. M., ‘The SQ-universality of some finitely presented groups’, J. Aust. Math. Soc. 16 (1973), 1–6.Google Scholar
Osin, D., ‘Acylindrically hyperbolic groups’, Trans. Amer. Math. Soc. 368(2) (2016), 851–888.Google Scholar
Pettet, M. R., ‘The automorphism group of a graph product of groups’, Comm. Algebra 27(10) (1999), 4691–4708.Google Scholar
Sale, A. and Susse, T., ‘Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian’, Preprint, 2017, arXiv:1706.07873.Google Scholar
Serre, J.-P., Trees, Springer Monographs in Mathematics (Springer, Berlin, 2003).Google Scholar
Tits, J., ‘Sur le groupe des automorphismes de certains groupes de Coxeter’, J. Algebra 113(2) (1988), 346–357.Google Scholar
Varghese, O., ‘The automorphism group of the universal Coxeter group’, Preprint, 2018.Google Scholar
Watatani, Y., ‘Property (T) of Kazhdan implies property (FA) of Serre’, Math. Japonica 27 (1982), 97–103.Google Scholar