Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T14:23:22.812Z Has data issue: false hasContentIssue false
  • English
  • Français

On commensurability of right-angled Artin groups II: RAAGs defined by paths

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

Published online by Cambridge University Press:  12 December 2019

MONTSERRAT CASALS–RUIZ ,
ILYA KAZACHKOV  and
ALEXANDER ZAKHAROV
MONTSERRAT CASALS–RUIZ
Affiliation:
Ikerbasque - Basque Foundation for Science and Matematika Saila, UPV/EHU, Sarriena s/n, 48940, Leioa - Bizkaia, Spain. e-mails: [email protected], [email protected]
ILYA KAZACHKOV
Affiliation:
Ikerbasque - Basque Foundation for Science and Matematika Saila, UPV/EHU, Sarriena s/n, 48940, Leioa - Bizkaia, Spain. e-mails: [email protected], [email protected]
ALEXANDER ZAKHAROV
Affiliation:
Chebyshev Laboratory, St Petersburg State University, 14th Line 29B, Vasilyevsky Island, 199178, St.Petersburg, Russia The Russian Foreign Trade Academy, 4a Pudovkina street, 119285, Moscow, Russia. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length n > 4 is commensurable to a RAAG defined by a tree of diameter 4 if and only if n ≡ 2 (mod 4). These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.

MSC classification

Primary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Secondary: 20F10: Word problems, other decision problems, connections with logic and automata 20F65: Geometric group theory 20E99: None of the above, but in this section
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 559 - 608
Copyright
© Cambridge Philosophical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Agol, I.. The virtual Haken conjecture. Doc. Math. 18: 1045–1087.Google Scholar
Bass, H.. The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. 3, 4 (1972), 603–614.CrossRefGoogle Scholar
Behrstock, J., Januszkiewicz, T. and Neumann, W.. Commensurability and QI classification of free products of finitely generated abelian groups. Proc. Amer. Math. Soc. 137, 3 (2009), 811–813.CrossRefGoogle Scholar
Behrstock, J. and Neumann, W.. Quasi-isometric classification of graph manifold groups. Duke Math. J. 141, 2 (2008), 217–240.CrossRefGoogle Scholar
Burger, M. and Mozes, S.. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92 (2000), 151–194.CrossRefGoogle Scholar
Casals-Ruiz, M., Kazachkov, I. and Zakharov, A.. On commensurability of right-angled Artin groups I: RAAGs defined by trees of diameter 4, Revista Matemática Iberoamericana 35:2 (2019), 521–560.CrossRefGoogle Scholar
Chistikov, D. and Haase, C.. On the complexity of quantified integer programming. In ICALP’17, vol. 80 of LIPIcs (2017), pp 94:1–94:13.Google Scholar
Deligne, P. and Mostow, G. D.. Commensurabilities among lattices in PU(1,n). Annals of Math. Stud, 132 (Princeton University Press, 1993).Google Scholar
Garrido, A.. Abstract commensurability and the Gupta–Sidki group, Groups Geom. Dyn. 10 (2016), 523–543.CrossRefGoogle Scholar
Grigorchuk, R. and Wilson, J., A structural property concerning abstract commensurability of subgroups. J. London Math. Soc. 68, 3 (2003), 671–682.CrossRefGoogle Scholar
Gromov, M.. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–78.CrossRefGoogle Scholar
Gromov, M.. Geometric group theory, vol. 2: asymptotic invariants of infinite groups. London Math. Soc. Lecture Notes 182 (1993), 1–295.Google Scholar
Huang, J.. Commensurability of groups quasi-isometric to RAAGs. Invent. Math. 213, 3 (2018), 1179–1247.CrossRefGoogle Scholar
Karrass, A., Pietrowski, A. and Solitar, D.. Finite and infinite cyclic extensions of free groups, J. Australian Math. Soc 16, 04 (1973), 458–466.CrossRefGoogle Scholar
Kim, S. and Koberda, T.. Embedability between right-angled Artin groups, Geom. Topol. 17 (2013), 493–530.CrossRefGoogle Scholar
Kim, S. and Koberda, T.. The geometry of the curve graph of a right-angled Artin group, Int. J. Algebra Comput. 24 (2014), 121–169.Google Scholar
Margulis, G. A.. Arithmeticity of nonuniform lattices, Funkcional. Anal. i Prilozen. 7, 3 (1973), 88–89.Google Scholar
Schwartz, R. E.. The quasi-isometry classification of rank one lattices, Publ. Math. Inst. Hautes Etudes Sci. 82 (1995), 133–168.CrossRefGoogle Scholar
Serre, J.-P.. Trees, (Springer-Verlag, 1980).CrossRefGoogle Scholar
Siegel, C. L.. Symplectic geometry, Amer. J. Math. 65 (1943), 1–86.CrossRefGoogle Scholar
Stallings, J. R.. On torsion-free groups with infinitely many ends, Ann. Math. (1968), 312-334.CrossRefGoogle Scholar
Weichsel, P., The Kronecker product of graphs. Proc. Amer. Math. Soc. 13 (1962), 47–52.CrossRefGoogle Scholar
Whyte, K.. Coarse bundles, arXiv:1006.3347.Google Scholar
Wise, D.. Non-Positively Curved Squared Complexes, Aperiodic Tilings and Non-Residually Finite Groups (Princeton University, 1996).Google Scholar