A converse to Moore's theorem on cellular automata

We prove a converse to Moore’s “Garden-of-Eden” theorem: a group G is amenable if and only if all cellular automata living on G that admit mutually erasable patterns also admit gardens of Eden. It had already been conjectured in [11; 1, Conjecture 6.2] that amenability could be characterized by cellular automata. We prove the first part of that conjecture.