## Electronic Communications in Probability

### Indicable groups and $p_c<1$

#### Abstract

A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of $\mathbb{Z}$ has a non-trivial percolation phase. Our main results prove this conjecture for certain groups, and in particular prove that any group with a non-trivial homomorphism into the additive group of real numbers satisfies the conjecture. We use this to reduce the conjecture to the case of hereditary just-infinite groups.

The novelty here is mainly in the methods used, combining the methods of EIT and evolving sets, and using the algebraic properties of the group to apply these methods.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 13, 10 pp.

Dates
Accepted: 27 December 2016
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.ecp/1485831618

Digital Object Identifier
doi:10.1214/16-ECP40

Mathematical Reviews number (MathSciNet)
MR3607808

Zentralblatt MATH identifier
1360.82037

#### Citation

Raoufi, Aran; Yadin, Ariel. Indicable groups and $p_c&lt;1$. Electron. Commun. Probab. 22 (2017), paper no. 13, 10 pp. doi:10.1214/16-ECP40. https://projecteuclid.org/euclid.ecp/1485831618

#### References

• [1] R.G. Alves, A. Procacci, R. Sanchis. Percolation on infinite graphs and isoperimetric inequalities. J. Stat. Phys. 149 (2012), 831–845.
• [2] O. Angel, I. Benjamini, N. Berger, Y. Peres. Transience of percolation clusters on wedges. EJP 11 (2006), 655–669.
• [3] E. Babson, I. Benjamini. Cut sets and normed cohomology with applications to percolation. Proc. Am. Math. Soc. 127 (1999), 589–597.
• [4] L. Bartholdi, R.I. Grigorchuk, Z. Šuniḱ. Branch groups. (Handbook of algebra Vol. 3 (2003), 989–1112. North-Holland, Amsterdam.) arXiv:math/0510294
• [5] H. Bass. The degree of polynomial growth of finitely generated nilpotent groups. Proceedings of the London Mathematical Society 25 (1972), 603–614.
• [6] I. Benjamini, R. Pemantle, Y. Peres. Unpredictable paths and percolation. Ann. of Probab. 26 (1998), 1198–1211.
• [7] I. Benjamini, O. Schramm. Percolation beyond $\mathbb{Z} ^d$, many questions and a few answers. Elect. Comm. in Probab. 1 (1996), 71–82.
• [8] B. Bollobas, O. Riordan. Percolation. (2006), Cambridge University Press.
• [9] E. Breuillard. On uniform exponential growth for solvable groups. arXiv:math/0602076
• [10] S.R. Broadbent, J.M. Hammersley. Percolation processes. Mathematical Proceedings of the Cambridge Philosophical Society. 53 (1957), Cambridge University Press.
• [11] E. Candellero, A. Teixeira. Percolation and isoperimetry on roughly transitive graphs. arXiv:1507.07765
• [12] T. Coulhon, L. Saloff-Coste. Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana 9 (1993), 293–314.
• [13] A. Dembo, R. Huang, B. Morris, Y. Peres. Transience in growing subgraphs via evolving sets. To appear in Ann. Inst. Henri Poincare.
• [14] Y. Guivarc’h. Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France. 101 (1973), 333–379.
• [15] R. Grigorchuk. On the Gap Conjecture concerning group growth. Bulletin of Mathematical Sciences 4 (2014), 113–128.
• [16] G.R. Grimmett. Percolation. (2010), (grundlehren der mathematischen wissenschaften).
• [17] M. Gromov. Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits). Pub. Math. de l’IHÉS 53 (1981), 53–78.
• [18] B. Kleiner. A new proof of Gromov’s theorem on groups of polynomial growth. Journal of the AMS 23 (2010), 815–829.
• [19] R. Lyons. Random walks and the growth of groups. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1361–1366.
• [20] R. Lyons, Y. Peres. Probability on trees and networks. (2016), Cambridge University Press. Available at http://pages.iu.edu/~rdlyons/
• [21] R. Lyons, O. Schramm. Indistinguishability of Percolation Clusters Ann. Probab. 27 (1999), 1809–1836.
• [22] J. Milnor. Growth of finitely generated solvable groups. J. Differential Geometry 2 (1968), 447–449.
• [23] B. Morris, Y. Peres. Evolving sets, mixing and heat kernel bounds. Prob. Th. Rel. Fields 133 (2005), 245–266.
• [24] R. Muchnik, I. Pak. Percolation on Grigorchuk groups. Communications in Algebra 29 (2001), 661–671.
• [25] V. Nekrashevych. Palindromic subshifts and simple periodic groups of intermediate growth. arXiv:1601.01033
• [26] N. Ozawa. A functional analysis proof of Gromov’s polynomial growth theorem. Ann. Sci. Ec. Norm. Super., to appear. arXiv:1510.04223
• [27] R. Pemantle, Y. Peres. On which graphs are all random walks in random environments transient? in: Random Discrete Structures (1996) Springer New York. p. 207–211.
• [28] G. Pete. Probability and geometry on groups. Available at: http://math.bme.hu/~gabor/
• [29] S. Rosset. A property of groups of non-exponential growth. Proceedings of the AMS 54 (1976), 24–26.
• [30] A. Teixeira. Percolation and local isoperimetric inequalities. Probability Theory and Related Fields, to appear.
• [31] J.A. Wolf. Growth of finitely generated solvable groups and curvature of Riemanniann manifolds. J. Differential Geometry 2 (1968) 421–446.