JournalsggdVol. 11, No. 4pp. 1307–1345

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
Asymptotic shapes for ergodic families of metrics on Nilpotent groups cover

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Abstract

Let Γ\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 Γ\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 Γ\Gamma; (iii) the sub-additive ergodic theorem over a general ergodic Γ\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 Γ\Gamma.

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