Mapping class groups have unique asymptotic cones

Mark Hagen (Bristol)

Abstract: The class of simplicial trees generalises in a "coarse" way, to the class of Gromov-hyperbolic spaces, in a "fine" way to real trees, and in a "high-dimensional" way to CAT(0) cube complexes. On the coarse side, cube complexes generalise to Bowditch's coarse median spaces, and on the fine side, to median spaces. Coarse median spaces are very general, and many examples from nature, like mapping class group of surfaces and fundamental groups of compact special cube complexes, have a stronger property, called hierarchical hyperbolicity. We introduce a fine-geometric notion, that of a "real cubing", that relates to cube complexes and median spaces in the same way that hierarchically hyperbolic spaces relate to cube complexes and coarse median spaces. One often passes from the coarse side to the fine side by taking asymptotic cones, and our first result is indeed that any asymptotic cone of a hierarchically hyperbolic group is bilipschitz homeomorphic to a real cubing. This generalises and strengthens a result of Behrstock-Drutu-Sapir about mapping class groups (and generalises the fact that asymptotic cones of hyperbolic groups are real trees). Using the real cubing structure, we show that for hierarchically hyperbolic groups with nice enough algebraic properties --- including mapping class groups and right-angled Artin groups --- any two asymptotic cones are bilipschitz equivalent. This is all joint work in progress with Montse Casals-Ruiz and Ilya Kazachkov.