Groups, Geometry, and Dynamics


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Volume 11, Issue 4, 2017, pp. 1307–1345
DOI: 10.4171/GGD/430

Published online: 2017-12-07

Asymptotic shapes for ergodic families of metrics on Nilpotent groups

Michael Cantrell[1] and Alex Furman[2]

(1) University of Illinois at Chicago, USA
(2) University of Illinois at Chicago, USA

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 $\Gamma$.

Keywords: Subadditive ergodic theorem, Carnot group, first passage percolation

Cantrell Michael, Furman Alex: Asymptotic shapes for ergodic families of metrics on Nilpotent groups. Groups Geom. Dyn. 11 (2017), 1307-1345. doi: 10.4171/GGD/430