For hierarchically hyperbolic groups, the boundary is connected if and only if the group is one-ended, and free product boundaries are characterized by factor boundaries.
An EZ-structure for the mapping class group
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abstract
We construct a boundary for the mapping class group Mod(S) of a surface S of finite type. The action of Mod(S) on this boundary is minimal, strongly proximal and topologically free. The boundary is the boundary of an EZ-structure for Mod(S).
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On the connectedness of the boundary of hierarchically hyperbolic spaces
For hierarchically hyperbolic groups, the boundary is connected if and only if the group is one-ended, and free product boundaries are characterized by factor boundaries.