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Title: Action rigidity of free products of hyperbolic manifold groups

Abstract:

Gromov's program for understanding finitely generated groups up to their large scale geometry considers three possible relations: quasi-isometry, abstract commensurability, and acting geometrically on the same proper geodesic metric space. A common model geometry for groups G and G′ is a proper geodesic metric space on which G and G′ act geometrically. A group G is action rigid if any group G′ that has a common model geometry with G is abstractly commensurable to G. We show that free products of closed hyperbolic manifold groups are action rigid. As a corollary, we obtain torsion-free, Gromov hyperbolic groups that are quasi-isometric, but do not even virtually act on the same proper geodesic metric space. This is joint work with Emily Stark.