Asymptotic shapes for ergodic families of metrics on Nilpotent groups
Michael Cantrell
University of Illinois at Chicago, USAAlex Furman
University of Illinois at Chicago, USA
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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