CAT(0) cube complexes and asymptotically rigid mapping class groups

In this article we study the asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. We present a family of CAT(0) cube complexes on which the latter groups act. Along the way, we determine in which cases the cube complexes introduced earlier by Genevois, Lonjou and Urech are CAT(0).