Gromov's Theorem

Gromov's Theorem states that any finitely generated group with a property called polynomial growth—meaning the number of distinct elements reachable within a certain "distance" grows at most as a polynomial function—must be virtually nilpotent. In simpler terms, such groups have a highly structured, almost "layered" nature, closely resembling groups composed of matrices that become increasingly "simple" at large scales. This theorem links geometric properties (growth rates) with algebraic structure, revealing that groups growing slowly are fundamentally close to well-understood, "tame" groups like those with upper triangular matrices.