Asymptotic Shapes for Ergodic Families of Metrics on Nilpotent Groups

TL;DR

This paper investigates the asymptotic geometric structures of ergodic metric families on finitely generated nilpotent groups, extending known theorems and analyzing limit shapes in percolation and ergodic contexts.

Contribution

It generalizes Pansu's theorem to ergodic families of metrics, studies limit shapes in first passage percolation with ergodic processes, and establishes a sub-additive ergodic theorem for nilpotent groups.

Findings

Asymptotic cones are described by Carnot-Caratheodory metrics.

Limit shapes for percolation are characterized in the nilpotent setting.

A sub-additive ergodic theorem is proven for ergodic actions on nilpotent groups.

Abstract

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