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

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