### Asymptotic dimension of minor-closed families, and beyond.

Combinatorics Seminar

24th February 2021, 4:00 pm – 5:00 pm

Virtual (online) Zoom seminar; a link will be sent to the Bristol Combinatorics Seminar and Analysis & Geometry Seminar mailing lists, the week before the seminar.

We consider the asymptotic dimension of the metric spaces given by the

shortest path metric of (edge-)weighted graphs. We prove that the

asymptotic dimension of any minor-closed family of weighted graphs, any

class of weighted graphs of bounded tree-width, and any class of graphs

of bounded layered tree-width are at most 2, 1, and 2, respectively.

The first result solves a question of Fujiwara and Papasoglu; the second

and third results solve a number of questions of Bonamy, Bousquet,

Esperet, Groenland, Pirot and Scott. These bounds for asymptotic

dimension are optimal and generalize and improve some results in the

literature, including results for Riemannian surfaces and Cayley graphs

of groups with a forbidden minor.

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