Property (T) (respectively aTmenability) is equivalent to admitting only a trivial action (respectively, a proper action) on a median space, and is also equivalent to admitting only a trivial action (respectively, a proper action) on a Hilbert space (so some L2). For p>2 I will investigate an analogous equivalent characterisation.

The mapping class group a solid handlebody of genus g sits between mapping class groups of surfaces and Out(F_n), in the sense there is an injective map to the mapping class group of the boundary and a surjective map to Out(F_g) via the action on the fundamental group. Similar behaviour happens with actions on associated spaces, such curve complexes and Teichmuller space. I’ll give an expository talk on this, partly in the context of our proof with Petersen that handlebody groups are virtual duality groups, and partly in the context of a problem list on handlebody groups written with Andrew, Hensel, and Hughes.