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Special cube complexes revisited: a quasi-median generalization

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  11 April 2022

Anthony Genevois
Anthony Genevois*
Affiliation:
Montpellier Alexander Grothendieck Institute, Université de Montpellier, Place Eugène Bataillon, 34090 Montpellier, France
*
e-mail: anthony.genevois@umontpellier.fr
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Abstract

In this article, we generalize Haglund and Wise’s theory of special cube complexes to groups acting on quasi-median graphs. More precisely, we define special actions on quasi-median graphs, and we show that a group which acts specially on a quasi-median graph with finitely many orbits of vertices must embed as a virtual retract into a graph product of finite extensions of clique-stabilizers. In the second part of the article, we apply the theory to fundamental groups of some graphs of groups called right-angled graphs of groups.

Keywords

quasi-median graphsgraph products of groupsspecial cube complexes

MSC classification

Primary: 20F65: Geometric group theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 3 , June 2023 , pp. 743 - 777
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work was supported by a public grant as part of the Fondation Mathématique Jacques Hadamard.

References

Agol, I., The virtual Haken conjecture . Doc. Math. 18(2013), 1045–1087, with an appendix by I. Agol, D. Groves, and J. Manning.CrossRefGoogle Scholar
Alonso, J., Inégalités isopérimétriques et quasi-isométries . C. R. Acad. Sci. Paris Sér. I Math. 311(1990), 761–764.Google Scholar
Alonso, J., Finiteness conditions on groups and quasi-isometries . J. Pure Appl. Algebra 95(1994), 121–129.CrossRefGoogle Scholar
Alonso, J., Dehn functions and finiteness properties of graph products . J. Pure Appl. Algebra 107(1996), 9–17.CrossRefGoogle Scholar
Antolín, Y. and Dreesen, D., The Haagerup property is stable under graph products. Preprint, 2014. arXiv:1305.6748Google Scholar
Antolín, Y. and Minasyan, A., Tits alternatives for graph products . J. Reine Angew. Math. 704(2015), 55–83.CrossRefGoogle Scholar
Bandelt, H.-J., Mulder, H. M., and Wilkeit, E., Quasi-median graphs and algebras . J. Graph Theory 18(1994), no. 7, 681–703.CrossRefGoogle Scholar
Bass, H., Covering theory for graphs of groups . J. Pure Appl. Algebra 89(1993), nos. 1–2, 3–47.CrossRefGoogle Scholar
Baudisch, A., Subgroups of semifree groups . Acta Math. Acad. Sci. Hungar. 38(1981), nos. 1–4, 19–28.CrossRefGoogle Scholar
Berlai, F. and de la Nuez González, J., Linearity of graph products. Preprint, 2019. arXiv:1906.11958 Google Scholar
Berlai, F. and Ferov, M., Separating cyclic subgroups in graph products of groups . J. Algebra 531(2019), 19–56.CrossRefGoogle Scholar
Bourdon, M., Immeubles hyperboliques, dimension conforme et rigidité de Mostow . Geom. Funct. Anal. 7(1997), no. 2, 245–268.CrossRefGoogle Scholar
Brešar, B., Chalopin, J., Chepoi, V., Gologranc, T., and Osajda, D., Bucolic complexes . Adv. Math. 243(2013), 127–167.CrossRefGoogle Scholar
Brown, R., Topology and groupoids, BookSurge, LLC, Charleston, SC, 2006, Third edition of it elements of modern topology (McGraw-Hill, New York, 1968), with 1 CD-ROM (Windows, Macintosh, and UNIX).Google Scholar
Burns, R., Karrass, A., and Solitar, D., A note on groups with separable finitely generated subgroups . Bull. Aust. Math. Soc. 36(1987), no. 1, 153–160.CrossRefGoogle Scholar
Carr, M., Two-generator subgroups of right-angled Artin groups are quasi-isometrically embedded. Preprint, 2015. arXiv:1412.0642 Google Scholar
Chepoi, V., Classification of graphs by means of metric triangles . Metody Diskret. Analiz. 49(1989), no. 96, 75–93.Google Scholar
Chepoi, V., Graphs of some $CAT(0)$ complexes. Adv. in Appl. Math. 24(2000), no. 2, 125–179.CrossRefGoogle Scholar
Cohen, D., Projective resolutions for graph products . Proc. Edinb. Math. Soc. (2) 38(1995), no. 1, 185–188.CrossRefGoogle Scholar
Cohen, D., Isoperimetric functions for graph products . J. Pure Appl. Algebra 101(1995), no. 3, 305–311.CrossRefGoogle Scholar
Crisp, J. and Wiest, B., Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups . Algebr. Geom. Topol. 4(2004), no. 1, 439–472.CrossRefGoogle Scholar
Davis, M. and Januszkiewicz, T., Right-angled Artin groups are commensurable with right-angled Coxeter groups . J. Pure Appl. Algebra 153(2000), no. 3, 229–235.CrossRefGoogle Scholar
Droms, C., Graph groups. Ph.D. thesis, Syracuse University, 1983.Google Scholar
Druţu, C. and Sapir, M., Tree-graded spaces and asymptotic cones of groups . Topology 44(2005), no. 5, 959–1058, with an appendix by D. Osin and M. Sapir.CrossRefGoogle Scholar
Duchamp, G. and Krob, D., The lower central series of the free partially commutative group . Semigroup Forum 45(1992), no. 3, 385–394.CrossRefGoogle Scholar
Duchamp, G. and Thibon, J.-Y., Simple orderings for free partially commutative groups . Internat. J. Algebra Comput. 2(1992), no. 3, 351–355.CrossRefGoogle Scholar
Ferov, M., On conjugacy separability of graph products of groups . J. Algebra 447(2016), 135–182.CrossRefGoogle Scholar
Genevois, A., Cubical-like geometry of quasi-median graphs and applications in geometric group theory. Ph.D. thesis, 2017. Preprint, arXiv:1712.01618 Google Scholar
Genevois, A., Hyperplanes of Squier’s cube complexes . Algebr. Geom. Topol. 18(2018), no. 6, 3205–3256.CrossRefGoogle Scholar
Genevois, A., Quasi-isometrically rigid subgroups in right-angled Coxeter groups. Algebraic & Geometric Topology. Preprint, 2019. arXiv:1909.04318 Google Scholar
Genevois, A., Embeddings into Thompson’s groups from quasi-median geometry . Groups Geom. Dyn. 13(2019), no. 4, 1457–1510.CrossRefGoogle Scholar
Genevois, A., On the geometry of van Kampen diagrams of graph products of groups. Preprint, 2019. arXiv:1901.04538 Google Scholar
Genevois, A., Negative curvature in graph braid groups . Internat. J. Algebra Comput. 31(2021), no. 1, 81–116.CrossRefGoogle Scholar
Genevois, A., Negative curvature of automorphism groups of graph products with applications to right-angled Artin groups. Automorphisms of graph products of groups and acylindrical hyperbolicity. Preprint, 2022. arXiv:1807.00622 Google Scholar
Genevois, A. and Martin, A., Automorphisms of graph products of groups from a geometric perspective . Proc. Lond. Math. Soc. (3) 119(2019), no. 6, 1745–1779.CrossRefGoogle Scholar
Green, E., Graph products of groups. Ph.D. thesis, University of Leeds, 1990.Google Scholar
Guba, V. and Sapir, M., On subgroups of the R. Thompson group $F$ and other diagram groups. Mat. Sb. 190(1999), no. 8, 3–60.Google Scholar
Hagen, M. and Przytycki, P., Cocompactly cubulated graph manifolds . Israel J. Math. 207(2015), no. 1, 377–394.CrossRefGoogle Scholar
Haglund, F., Isometries of CAT(0) cube complexes are semi-simple. Preprint, 2007. arXiv:0705.3386 Google Scholar
Haglund, F. and Wise, D., Special cube complexes . Geom. Funct. Anal. 17(2008), no. 5, 1551–1620.CrossRefGoogle Scholar
Haglund, F. and Wise, D., Coxeter groups are virtually special . Adv. Math. 224(2010), no. 5, 1890–1903.CrossRefGoogle Scholar
Higgins, P., The fundamental groupoid of a graph of groups . J. Lond. Math. Soc. (2) s2–13(1976), no. 1, 145–149.CrossRefGoogle Scholar
Hsu, T. and Wise, D., On linear and residual properties of graph products . Michigan Math. J. 46(1999), 251–259.CrossRefGoogle Scholar
Humphries, S., On representations of Artin groups and the Tits conjecture . J. Algebra 169(1994), no. 3, 847–862.CrossRefGoogle Scholar
Metaftsis, V. and Raptis, E., On the profinite topology of right-angled Artin groups . J. Algebra 320(2008), no. 3, 1174–1181.CrossRefGoogle Scholar
Minasyan, A., Hereditary conjugacy separability of right-angled Artin groups and its applications . Groups Geom. Dyn. 6(2012), no. 2, 335–388.CrossRefGoogle Scholar
Reckwerdt, E., Weak amenability is stable under graph products . J. Lond. Math. Soc. (2) 96(2017), no. 1, 133–155.CrossRefGoogle Scholar
Scott, P. and Wall, T., Topological methods in group theory, Homological group theory (Proc. Sympos., Durham) . London Math. Soc. Lecture Note Ser. 36(1977), no. 1979, 137–203.Google Scholar
Serre, J.-P., Trees, Springer Monographs in Mathematics, Springer, Berlin, 2003, translated from the French original by J. Stillwell, corrected 2nd printing of the 1980 English translation.Google Scholar
Tidmore, J., Cocompact cubulations of mixed 3-manifolds . Groups Geom. Dyn. 12(2018), no. 4, 1429–1460.CrossRefGoogle Scholar
Wade, R., The lower central series of a right-angled Artin group . Enseign. Math. 61(2015), nos. 3–4, 343–371.CrossRefGoogle Scholar