In their paper, Branman, Domat, Hoganson and Lymann proved that if a topological group acts in a “nice” way on a simplicial graph, then the group has a well defined geometry that makes it quasi-isometric to the graph. These actions generalize a Svârc-Milnor action to the context of coarsely-boundedly (CB) generated Polish groups.

We adapt these ideas to the context of locally bounded Polish groups and then construct an arc and curve model coarsely equivalent to Map(S) when Map(S) is locally bounded. We then use this model to show that the asymptotic dimension of Map(S) is infinite when S has a non-displaceable subsurface.