Hyperbolicity and uniformly Lipschitz affine actions on subspaces of \(L^1\). (English) Zbl 1536.22017
The main result is the following theorem: every hyperbolic group \(\Gamma\) admits a proper affine uniformly Lipschitz action on a subspace of an \(L^1\) space. The strong point and the improvement of the previous result of the author, is that this theorem is valid in general case, and not only residually finite. One should note though that it is unknown whether every hyperbolic group is residually finite.
The \(L^1\) space in question arises as a direct sum of \(\ell^1(\Gamma)\) and a quite complicated \(L^1\)-space containing an isometric copy of a Hilbert space.
This should be compared with a result of C. Drutu and J. M. Mackay [“Actions of acylindrically hyperbolic groups on \(\ell^1\)”, Preprint, arXiv:2309.12915] proving that every residually finite hyperbolic group admits a uniformly Lipschitz affine action on \(\ell^1(\mathbb N)\) with quasi-isometrically embedded (and thus proper) orbits.
The \(L^1\) space in question arises as a direct sum of \(\ell^1(\Gamma)\) and a quite complicated \(L^1\)-space containing an isometric copy of a Hilbert space.
This should be compared with a result of C. Drutu and J. M. Mackay [“Actions of acylindrically hyperbolic groups on \(\ell^1\)”, Preprint, arXiv:2309.12915] proving that every residually finite hyperbolic group admits a uniformly Lipschitz affine action on \(\ell^1(\mathbb N)\) with quasi-isometrically embedded (and thus proper) orbits.
Reviewer: Yulia Kuznetsova (Besançon)
MSC:
| 22D55 | Kazhdan’s property (T), the Haagerup property, and generalizations |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 22D12 | Other representations of locally compact groups |
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