Asymptotic shapes for ergodic families of metrics on Nilpotent groups

Abstract

Let $\Gamma$ be a finitely generated virtually nilpotent group. We consider three closely related problems: (i) convergence to a deterministic asymptotic cone for an equivariant ergodic family of inner metrics on $\Gamma$, generalizing Pansu's theorem; (ii) the asymptotic shape theorem for first passage percolation for general (not necessarily independent) ergodic processes on edges of a Cayley graph of $\Gamma$; (iii) the sub-additive ergodic theorem over a general ergodic $\Gamma$-action. The limiting objects are given in terms of a Carnot–Carathéodory metric on the graded nilpotent group associated to the Mal'cev completion of $\Gamma$.