Benjamini–Nekrashevych–Pete Conjecture on Strongly Scale-Invariant Groups

Prove that every finitely generated group G which admits a strongly scale-invariant monomorphism f: G → G—meaning f is injective, f(G) is a proper finite-index subgroup of G, and the intersection ⋂_{k≥0} f^k(G) is finite—is virtually nilpotent.

Background

The notion of scale-invariant groups was introduced by Benjamini in connection with renormalization in percolation. A conjecture that all finitely generated scale-invariant groups have polynomial growth was later shown to be false by Nekrashevych and Pete, which motivated a stronger property termed strong scale-invariance.

In response, the Benjamini–Nekrashevych–Pete conjecture asserts that finitely generated strongly scale-invariant groups are virtually nilpotent. This conjecture has been proved under additional hypotheses (e.g., normality of the subgroup chain, finiteness of the discriminant group, and for virtually polycyclic groups). The present paper provides an elementary proof in the specific case of semi-direct products Z^n ⋊_A Z, further supporting the conjecture but not resolving it in full generality.