Transitive graphs uniquely determined by their local structure

Frisch, Joshua; Tamuz, Omer

doi:10.1090/proc/12901

Transitive graphs uniquely determined by their local structure
HTML articles powered by AMS MathViewer

by Joshua Frisch and Omer Tamuz

Proc. Amer. Math. Soc. 144 (2016), 1913-1918

DOI: https://doi.org/10.1090/proc/12901

Published electronically: October 1, 2015

PDF | Request permission

Abstract:

We give an example of an infinite, vertex transitive graph that has the following property: it is the unique completion to a transitive graph of a large enough finite subgraph of itself.

References

David Aldous and Russell Lyons, Processes on unimodular random networks, Electron. J. Probab. 12 (2007), no. 54, 1454–1508. MR 2354165, DOI 10.1214/EJP.v12-463
David Aldous and J. Michael Steele, The objective method: probabilistic combinatorial optimization and local weak convergence, Probability on discrete structures, Encyclopaedia Math. Sci., vol. 110, Springer, Berlin, 2004, pp. 1–72. MR 2023650, DOI 10.1007/978-3-662-09444-0_{1}
Itai Benjamini and Nicolas Curien, Ergodic theory on stationary random graphs, Electron. J. Probab. 17 (2012), no. 93, 20. MR 2994841, DOI 10.1214/EJP.v17-2401
Itai Benjamini and Oded Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), no. 23, 13. MR 1873300, DOI 10.1214/EJP.v6-96
Christophe Champetier, L’espace des groupes de type fini, Topology 39 (2000), no. 4, 657–680 (French, with English summary). MR 1760424, DOI 10.1016/S0040-9383(98)00063-9
Yves Cornulier, On the Cantor-Bendixson rank of metabelian groups, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 2, 593–618 (English, with English and French summaries). MR 2895067, DOI 10.5802/aif.2623
Yves de Cornulier, Luc Guyot, and Wolfgang Pitsch, On the isolated points in the space of groups, J. Algebra 307 (2007), no. 1, 254–277. MR 2278053, DOI 10.1016/j.jalgebra.2006.02.012
Russell Lyons with Yuval Peres, Probability on tress and networks, Cambridge University Press, 2015. In preparation. Current version available at http://pages.iu.edu/\string rdlyons/.
Vladimir Trofimov, Automorphism groups of graphs as topological groups, Matematicheskie Zametki 38 (1985), no. 3, 378–385.