Speaker
Description
In topological dimension theory, a well known Hurewicz theorem for dimension-lowering maps states that if $f:X\to Y$ is a closed map of metric spaces, then
$\dim X \leq \dim Y + \dim (f)$, where $\dim(f) := \sup\ \{ \dim(f^{-1}(y))\ | \ y\in Y\}$. This theorem was extended to asymptotic dimension $\mathrm{asdim}$, and in particular to $\mathrm{asdim}$ of groups - in 2006, Dranishnikov and Smith proved a Hurewicz-type formula, which states that if $f:G\to H$ is a group homomorphism, then $\mathrm{asdim} \ G\leq \mathrm{asdim}\ H + \mathrm{asdim}\ \mathrm{ker}(f)$.
We will show that the analogous formula is true for countable approximate groups, i.e., we will present the proof of the following:
Theorem: Let $(\Xi, \Xi^\infty)$, $(\Lambda, \Lambda^\infty)$ be countable approximate groups and let $f: (\Xi, \Xi^\infty) \to (\Lambda, \Lambda^\infty)$ be a global morphism. Then
$ \mathrm{asdim}\ \Xi \leq \mathrm{asdim} \ \Lambda + \mathrm{asdim}\ ([\mathrm{ker}(f)]_c).$
References:
[1] N. Brodskiy, J. Dydak, M. Levin, and A. Mitra, A Hurewicz theorem for the Assouad-Nagata dimension, J. Lond. Math. Soc. (2), 77(3):741--756, 2008.
[2] M. Cordes, T. Hartnick and V. Tonić, Foundations of geometric approximate group theory, preprint, 2024, https://arxiv.org/pdf/2012.15303.pdf
[3] A. Dranishnikov and J. Smith, Asymptotic dimension of discrete groups,
Fund. Math. 189(1):27--34, 2006.
[4] T. Hartnick and V. Tonić, Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups, preprint, 2024.
