Skip to main content Accessibility help
×
Home
Hostname: page-component-768dbb666b-prhj4 Total loading time: 0.328 Render date: 2023-02-06T19:25:08.354Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Small non-Leighton two-complexes

Published online by Cambridge University Press:  05 September 2022

NATALIA S. DERGACHEVA
Affiliation:
Faculty of Mechanics and Mathematics of Moscow State University,Moscow 119991, Leninskie gory, MSU, Moscow, Russia. and Moscow Center for Fundamental and Applied Mathematics e-mails: nataliya.dergacheva@gmail.com, klyachko@mech.math.msu.su
ANTON A. KLYACHKO
Affiliation:
Faculty of Mechanics and Mathematics of Moscow State University,Moscow 119991, Leninskie gory, MSU, Moscow, Russia. and Moscow Center for Fundamental and Applied Mathematics e-mails: nataliya.dergacheva@gmail.com, klyachko@mech.math.msu.su

Abstract

How many 2-cells must two finite CW-complexes have to admit a common, but not finite common, covering? Leighton’s theorem says that both complexes must have 2-cells. We construct an almost (?) minimal example with two 2-cells in each complex.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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.)

Footnotes

This work was supported by the Russian Science Foundation, project no. 22-11-00075.

References

Abello, J., Fellows, M. R. and Stillwell, J. C.. On the complexity and combinatorics of covering finite complexes. Australas. J. Combin. 4 (1991), 103112.Google Scholar
Bass, H. and Kulkarni, R.. Uniform tree lattices. J. Amer. Math. Soc. 3:4 (1990), 843902.CrossRefGoogle Scholar
Bondarenko, I. and Kivva, B.. Automaton groups and complete square complexes. Groups, Geometry, and Dynamics, 16:1 (2022), 305332. See also arXiv: 707.00215CrossRefGoogle Scholar
Bridson, M. and Shepherd, S.. Leighton’s theorem: extensions, limitations, and quasitrees. Algebraic and Geometric Topology (to appear). See also arXiv: 009.04305.Google Scholar
Caprace, P.-E. and Wesolek, P.. Indicability, residual finiteness, and simple subquotients of groups acting on trees. Geometry and Topology 22:7 (2018), 41634204. See also arXiv: 708.04590CrossRefGoogle Scholar
Casals-Ruiz, M., Kazachkov, I. and Zakharov, A.. Commensurability of Baumslag–Solitar groups. Indiana Univ. Math. J. 70:6 (2021), 25272555. See also arXiv: 1910.02117CrossRefGoogle Scholar
Fomenko, A. and Fuchs, D.. Homotopical topology, 2nd ed., Graduate Texts in Math. vol. 273 (Springer, Cham, 2016).Google Scholar
Janzen, D. and Wise, D. T.. A smallest irreducible lattice in the product of trees. Algebraic and Geometric Topology 9:4 (2009), 21912201.CrossRefGoogle Scholar
Kargapolov, M. I. and Merzljakov, Yu. I.. Fundamentals of the theory of groups. Graduate Texts in Math. 62 (Springer, 1979).Google Scholar
Leighton, F. T.. Finite common coverings of graphs. J. Combin. Theory, Series B 33:3 (1982), 231238.CrossRefGoogle Scholar
Levitt, G.. Quotients and subgroups of Baumslag–Solitar groups. J. Group Theory, 18:1 (2015), 143. See also arXiv: 308.5122CrossRefGoogle Scholar
Lyndon, R. and Schupp, P.. Combinatorial Group Theory (Springer, 2015).Google Scholar
Meskin, S.. Nonresidually finite one-relator groups. Trans. Amer. Math. Soc., 164 (1972), 105114.CrossRefGoogle Scholar
Neumann, W. D.. On Leighton’s graph covering theorem. Groups, Geometry, and Dynamics, 4:4 (2010), 863872. See also arXiv: 906.2496CrossRefGoogle Scholar
Shepherd, S., Gardam, G. and Woodhouse, D. J.. Two generalisations of Leighton’s Theorem, arXiv:1908.00830.Google Scholar
Tucker, T. W.. Some topological graph theory for topologists: A sampler of covering space constructions. In: Latiolais P. (eds) Topology and Combinatorial Group Theory. Lecture Notes in Math., 1440 (Springer, Berlin, Heidelberg, 1990).Google Scholar
Wise, D. T.. Non-positively curved squared complexes: Aperiodic tilings and non-residually finite groups. PhD. thesis. Princeton Univeristy (1996).Google Scholar
Wise, D. T.. Complete square complexes. Comment. Math. Helv. 82:4 (2007), 683724.CrossRefGoogle Scholar
Woodhouse, D.. Revisiting Leighton’s theorem with the Haar measure. Math. Proc. Camb. Phil. Soc. 170:3 (2021), 615623. See also arXiv: 806.08196CrossRefGoogle Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Small non-Leighton two-complexes
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Small non-Leighton two-complexes
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Small non-Leighton two-complexes
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *