The critical point for percolation on groups

I will discuss a conjecture of Benjamini & Schramm from 1996: Any Cayley graph has a non-trivial critical point for percolation (i.e. p_c<1) unless the underlying group is a I finite extension of Z.

I will try to present a strategy to prove this conjecture (in fact some stronger form of it), that involves the notion of EIT = exponential intersection tail measures. Hopefully, all the notions involved (percolation, the critical point p_c, EIT, etc.) will be explained. The aim is to learn these notions and perhaps discuss the weakness or plausibility of the strategy proposed.