Asymptotic Shapes for Ergodic Families of Metrics on Nilpotent Groups

Michael Cantrell; Alex Furman

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arXiv · 1508.00244

Asymptotic Shapes for Ergodic Families of Metrics on Nilpotent Groups

Michael Cantrell·Alex Furman

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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.

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Michael Cantrell, Alex Furman. 2015-09-24. Asymptotic Shapes for Ergodic Families of Metrics on Nilpotent Groups. https://arxiv.org/abs/1508.00244

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