Action rigidity for free products of hyperbolic manifold groups
Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, that residually finite groups...
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| Médium: | Journal article |
| Jazyk: | English |
| Vydáno: |
Association des Annales de l'Institut Fourier
2023
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| Shrnutí: | Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, that residually finite groups that have a common model geometry are abstractly commensurable. Our result gives the first examples of hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. An important component of the proof is a generalization of Leighton’s graph covering theorem. The main theorem depends on residual finiteness, and we show that finite extensions of uniform lattices in rank-1 symmetric spaces that are not residually finite would give counterexamples. |
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