Graphs of curves on infinite-type surfaces with mapping class group actions

Durham, Matthew Gentry; Fanoni, Federica; Vlamis, Nicholas G.

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2581-2612.

Nous étudions le cas où le groupe modulaire d’une surface de type infini admet une action avec orbites non bornées sur un graphe connexe dont les sommets sont des courbes fermées simples de $S$. Nous définissons un invariant topologique pour surfaces de type infini qui détecte dans de nombreux cas s’il y a une telle action. Nous en déduissons que beaucoup de gros groupes modulaires, en tant que groupes topologiques non localement compacts, ont géométrie grossière non banale au sens de Rosendal.

We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.

Classification : 57S05, 37E30, 20F65, 57M07
Keywords: mapping class groups, surface homeomorphisms, curve graphs, infinite-type surfaces
Mot clés : groupes modulaires, homéomorphismes des surfaces, graphes des courbes, surfaces de type infini

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