In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct an interesting action of the mapping class group on a subgraph of the arc graph. This subgraph naturally emerges from a new characterisation of infinite-type surfaces in terms of homeomorphic subsurfaces.
[Bro10] On the structure of perfect sets of points, KNAW, Proceedings, Volume 12 (1910), pp. 785-794
[FHV19] Big mapping class groups acting on homology (2019) (https://arxiv.org/abs/1905.12509, to appear in Indiana University Mathematics Journal)
[LPRT19] Surfaces of infinite type, 2019 (Available at http://aimpl.org/genusinfinity)
[MR19] Large scale geometry of big mapping class groups (2019) (https://arxiv.org/abs/1912.10914)