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A note on morphisms to wreath products

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  08 October 2025

ANTHONY GENEVOIS  and
ROMAIN TESSERA
ANTHONY GENEVOIS
Affiliation:
Institut Montpellierain Alexander Grothendieck, 499-554 Rue du Truel, 34090 Montpellier, France. e-mail: anthony.genevois@umontpellier.fr
ROMAIN TESSERA
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche, Place Aurélie Nemours, 75013 Paris, France. e-mail: romain.tessera@imj-prg.fr
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Abstract

Given a morphism $\varphi \;:\; G \to A \wr B$ from a finitely presented group G to a wreath product $A \wr B$, we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free subgroup and $\varphi$ factors through an acylindrically hyperbolic quotient of G. As direct applications, we classify the finitely presented subgroups in $A \wr B$ up to isomorphism and we deduce that a finitely presented group having a wreath product $(\text{non-trivial}) \wr (\text{infinite})$ as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier–Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings.

MSC classification

Primary: 20E22: Extensions, wreath products, and other compositions 20F65: Geometric group theory
Secondary: 20E36: Automorphisms of infinite groups 20F28: Automorphism groups of groups 20F67: Hyperbolic groups and nonpositively curved groups

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 28
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Antolín, Y. and Minasyan, A.. Tits alternatives for graph products. J. Reine Angew. Math. 704 (2015), 55–83.Google Scholar
Baumslag, G.. Topics in combinatorial group theory. Lectures in Mathematics, ETH Zürich (Birkhäuser Verlag, Basel, 1993).10.1007/978-3-0348-8587-4CrossRefGoogle Scholar
Bachmuth, S., Baumslag, G., Dyer, J. and Mochizuki, H.. Automorphism groups of two generator metabelian groups. J. London Math. Soc. (2), 36(3) (1987), 393–406.Google Scholar
Benli, M., Grigorchuk, R. and de la Harpe, P.. Amenable groups without finitely presented amenable covers. Bull. Math. Sci. 3(1) (2013), 73–131.10.1007/s13373-013-0031-5CrossRefGoogle Scholar
Burns, R. and Hale, V.. A note on group rings of certain torsion-free groups. Canad. Math. Bull. 15 (1972), 441–445.10.4153/CMB-1972-080-3CrossRefGoogle Scholar
Bandelt, H.-J., Mulder, H.M. and Wilkeit, E.. Quasi-median graphs and algebras. J. Graph Theory 18(7) (1994), 681–703.10.1002/jgt.3190180705CrossRefGoogle Scholar
Chatterji, I., Druţu, C., and Haglund, F.. Kazhdan and Haagerup properties from the median viewpoint. Adv. Math. 225 (2010), 882–921.Google Scholar
Cannon, J., Floyd, W. and Parry, W.. Introductory notes on Richard Thompson’s groups. Enseign. Math. (2), 42(3-4) (1996), 215–256.Google Scholar
Champetier, C. and Guirardel, V.. Limit groups as limits of free groups. Israel J. Math. 146 (2005), 1–75.CrossRefGoogle Scholar
Chepoi, V.. Graphs of some $\rm CAT(0)$ complexes. Adv. in Appl. Math. 24(2) (2000), 125–179.Google Scholar
Cornulier, Y. and Kar, A.. On property (FA) for wreath products. J. Group Theory 14(1) (2011), 165–174.Google Scholar
Cornulier, Y.. Finitely presented wreath products and double coset decompositions. Geom. Dedicata 122 (2006), 89–108.10.1007/s10711-006-9061-4CrossRefGoogle Scholar
Cornulier, Y.. Group actions with commensurated subsets, wallings and cubings. Preprint ArXiv:1302.5982, (2013).Google Scholar
Coulon, R.. On the geometry of Burnside quotients of torsion free hyperbolic groups. Internat. J. Algebra Comput. 24(3) (2014), 251–345.10.1142/S0218196714500143CrossRefGoogle Scholar
Dahmani, F., Guirardel, V. and Osin, D.. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Mem. Amer. Math. Soc. 245(1156) (2017), v+152.Google Scholar
Gardam, G.. A counterexample to the unit conjecture for group rings. Preprint ArXiv:2102.11818, To appear in: Ann. of Math. (2021).10.4007/annals.2021.194.3.9CrossRefGoogle Scholar
Genevois, A.. Cubical-like geometry of quasi-median graphs and applications to geometric group theory. PhD. thesis. Aix-Marseille Université Arxiv:1712.01618, (2017).Google Scholar
Genevois, A.. Automorphisms of graph products of groups and acylindrical hyperbolicity. Preprint ArXiv:1807.00622, (2018).Google Scholar
Genevois, A.. Acylindrical hyperbolicity from actions on $\rm CAT(0)$ cube complexes: a few criteria. New York J. Math. 25 (2019), 1214–1239.Google Scholar
Genevois, A.. Hyperbolic and cubical rigidities of Thompson’s group V . J. Group Theory 22(2) (2019), 313–345.CrossRefGoogle Scholar
Genevois, A.. On the geometry of van Kampen diagrams of graph products of groups. Preprint: ArXiv:1901.04538, (2019).Google Scholar
Genevois, A. and Martin, A.. Automorphisms of graph products of groups from a geometric perspective. Proc. London Math. Soc. 119(6) (2019), 1745–1779.Google Scholar
Green, E.. Graph products of groups. PhD. thesis. University of Leeds (1990).Google Scholar
Genevois, A. and Tessera, R.. Asymptotic geometry of lamplighters over one-ended groups. Preprint: Arxiv:2105.04878, (2021).Google Scholar
Genevois, A. and Tessera, R.. Lamplighter-like geometry of groups. Preprint: ArXiv:2401.13520, (2024).Google Scholar
Mohammadi Hassanabadi, A.. Automorphisms of permutational wreath products. J. Austral. Math. Soc. (Series A) 26 (1978), 198–208.Google Scholar
Higman, G.. A finitely generated infinite simple group. J. London Math. Soc. 26 (1951), 61–64.Google Scholar
Houghton, C.H.. Wreath products of groupoids. J. London Math. Soc. 10(2) (1975), 179–188.Google Scholar
Kato, M.. On groups whose actions on finite-dimensional $\rm CAT(0)$ spaces have global fixed points. J. Group Theory 22(6) (2019), 1089–1099.10.1515/jgth-2018-0116CrossRefGoogle Scholar
Krasner, M. and Kaloujnine, L.. Produit complet des groupes de permutations et problème d’extension de groupes. III. Acta Sci. Math. (Szeged) 14 (1951), 69–82.Google Scholar
Lysënok, I.. A set of defining relations for the Grigorchuk group. Mat. Zametki 38(4) (1985), 503–516, 634.Google Scholar
Lyul’ko, A.. Normal subgroups of Abels groups. Mat. Zametki 36(3) (1984), 289–294.Google Scholar
Mal’cev, A.. Nilpotent torsion-free groups. Izv Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 201–212.Google Scholar
Minasyan, A. and Osin, D.. Acylindrical hyperbolicity of groups acting on trees. Math. Ann. 362(3) (2015), 1055–1105.10.1007/s00208-014-1138-zCrossRefGoogle Scholar
Matucci, F. and Silva, P.. Extensions of automorphisms of self-similar groups. J. Group Theory 24(5) (2021), 857–897.10.1515/jgth-2020-0034CrossRefGoogle Scholar
Neumann, P.. On the structure of standard wreath products of groups. Math. Z. 84 (1964), 343–373.Google Scholar
Osajda, D.. Group cubization. Duke Math. J. 167(6) (2018), 1049–1055. With an appendix by M. Pichot.Google Scholar
Osin, D.. Kazhdan constants of hyperbolic groups. Funktsional. Anal. i Prilozhen. 36(4) (2002), 46–54.Google Scholar
Osin, D.. Acylindrically hyperbolic groups. Trans. Amer. Math. Soc. 368 (2016), 851–888.10.1090/tran/6343CrossRefGoogle Scholar
Passman, D.. The Algebraic Structure of Group Rings. (Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1985). Reprint of the 1977 original.Google Scholar
Pauls, S.. The large scale geometry of nilpotent Lie groups. Comm. Anal. Geom. 9(5) (2001), 951–982.10.4310/CAG.2001.v9.n5.a2CrossRefGoogle Scholar
Roller, M.. Pocsets, median algebras and group actions; an extended study of dunwoody’s construction and sageev’s theorem. “Habilitationsschrift” at Universit\"{a}t Regensburg. ArXiv:1607.07747, (1998).Google Scholar
Sageev, M.. Ends of group pairs and non-positively curved cube complexes. Proc. London Math. Soc. (3) 71(3) (1995), 585–617.Google Scholar
Stein, M., Taback, J. and Wong, P.. Automorphisms of higher rank lamplighter groups. Internat. J. Algebra Comput. 25(8) (2015), 1275–1299.Google Scholar
Scott, P. and Wall, T.. Topological methods in group theory. In Homological Group Theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser. vol.36 (Cambridge University Press, Cambridge-New York, 1979), 137–203.Google Scholar