Totally geodesic subvarieties of moduli space have semisimple Deligne-Mumford boundary and are hierarchically hyperbolic.
Structure invariant properties of the hierarchically hyperbolic boundary
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We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant under a "maximization" procedure introduced by the first two authors and Durham.
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The geometry of totally geodesic subvarieties of moduli spaces of Riemann surfaces
Totally geodesic subvarieties of moduli space have semisimple Deligne-Mumford boundary and are hierarchically hyperbolic.