A finitely generated group $\Gamma$ is called strongly scale-invariant if there exists an injective endomorphism $\varphi: \Gamma \to \Gamma$ with the image $\varphi(\Gamma)$ of finite index in and subgroup $\bigcap\_{n>0}\varphi^n(\Gamma)$ finite. The only known examples such groups are virtually nilpotent, or equivalently, all have polynomial growth. question by Nekrashevych Pete asks whether...

We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: