## Electronic Communications in Probability

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

### 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**

Received: 30 June 2016

Accepted: 27 December 2016

First available in Project Euclid: 31 January 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/16-ECP40

**Mathematical Reviews number (MathSciNet)**

MR3607808

**Zentralblatt MATH identifier**

1360.82037

**Subjects**

Primary: 82B43: Percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B26: Phase transitions (general)

**Keywords**

percolation Cayley graphs phase transition

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

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