Asymptotic shapes for ergodic families of metrics on Nilpotent groups

  • Michael Cantrell

    University of Illinois at Chicago, USA
  • Alex Furman

    University of Illinois at Chicago, USA

Abstract

Let 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 , 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 ; (iii) the sub-additive ergodic theorem over a general ergodic -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 .

Cite this article

Michael Cantrell, Alex Furman, Asymptotic shapes for ergodic families of metrics on Nilpotent groups. Groups Geom. Dyn. 11 (2017), no. 4, pp. 1307–1345

DOI 10.4171/GGD/430