John Mackay

Asymptotic dimension for covers with controlled growth

The "asymptotic dimension" of a metric space is an invariant introduced
by Gromov as a large-scale analogue of topological dimension, where one
covers the space by uniformly bounded sets with controlled overlap. It
has many applications, one of which is as an obstruction to coarse
embeddings between spaces. I'll discuss recent work with Hume and
Tessera where we study analogous invariants where the covers are allowed
to be unbounded, but to grow slowly. This allowed us to show, for
example, that there is no coarse embedding of the hyperbolic plane into
the product of a regular tree and any Euclidean space, answering a
question of Benjamini-Schramm-Timar.