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Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations
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Relations between boundaries of groups acting on hyperbolic spaces with WPD elements
I think the equivariant embedding you ask for is given in Theorem 3.2 of this paper:
https://arxiv.org/pdf/1601.00101.pdf.
Actually, there is such an embedding any time you cone off uniformly quasic …