In their theorem from 2006, A. Dranishnikov and J. Smith proved that if $f:G\to H$ is a group homomorphism, then the following formula for asymptotic dimension is true: $\mathrm{asdim} G \leq \mathrm{asdim} H + \mathrm{asdim} (\mathrm{ker} f)$. This result is known as the Hurewicz-type formula, after a 1927 theorem from classical topological dimension theory by W. Hurewicz, which inspired it.

In this talk we will establish a similar formula to the one by Dranishnikov and Smith, for the following setup: whenever $(\Xi, \Xi^\infty)$ and $(\Lambda,\Lambda^\infty)$ are countable approximate groups and $f:(\Xi, \Xi^\infty) \to (\Lambda,\Lambda^\infty)$ is a (general) quasimorphism, i.e., a quasimorphism which need not be symmetric nor unital, then the following formula is true: \(\mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda + \mathrm{asdim} (f^{-1}(f(e_\Xi)D(f)^{-1}D(f))),\) where $D(f)$ is the defect set of the quasimorphism $f$. It follows as a corollary that if $f:G\to H$ is a quasimorphism of countable groups, then \(\mathrm{asdim} G\leq \mathrm{asdim} H + \mathrm{asdim} (f^{-1}(f(e_\Xi)D(f)^{-1}D(f))).\) In particular, whenever the quasimorphism $f$ is symmetric and unital, we can replace $f^{-1}(f(e_\Xi)D(f)^{-1}D(f))$ in the formulas above by $f^{-1}(D(f))$.