Accessibility and Twin-width

Abstract

We show that finite twin-width does not imply accessibility for finitely generated groups, which answers a question of Esperet. That is, we prove that there exists a finitely generated group $\Gamma$ that has finite uniform twin-width but is not accessible. In particular, for every finite generating set $S$ of $\Gamma$, the Cayley graph $Cay(\Gamma ; S)$ has finite twin-width but is not accessible. The example is obtained by combining Wilkes construction of a finitely generated inaccessible residually $p$-finite groups with a result of Bonnet, Geniet, Tessera and Thomasse regarding the twin-width of groups acting faithfully on regular rooted trees.