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approximation_properties_summary [2019/10/04 15:07] d.gromada [Concrete examples] |
approximation_properties_summary [2021/11/23 11:56] (current) |
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===== Mutual relationships ===== | ===== Mutual relationships ===== | ||
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+ | Residually finite ⇒ property (F) | ||
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+ | Amenable ⇒ Haagerup | ||
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+ | Property (T) and (F) ⇒ Residually finite | ||
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+ | Property (T) and Haagerup ⇔ finite | ||
===== Stability results ===== | ===== Stability results ===== | ||
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^[[residual_finiteness|Resid. finite]]| | | | | | | yes | | ^[[residual_finiteness|Resid. finite]]| | | | | | | yes | | ||
^[[kazhdan_property|Kazhdan (T)]] | | | | | | yes | | | ^[[kazhdan_property|Kazhdan (T)]] | | | | | | yes | | | ||
- | ^[[haagerup_property|Haagerup]] | | yes | | | | | | | + | ^[[haagerup_property|Haagerup]] | | yes | | | yes | | | |
- | ^[[amenability|Amenable]] | | no | | | yes | | | | + | ^[[amenability|Amenable]] | yes | no | | | yes | | | |
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^[[kazhdan_property|Kazhdan (T)]] | no | no | no | no | no | no | | no || | ^[[kazhdan_property|Kazhdan (T)]] | no | no | no | no | no | no | | no || | ||
^[[haagerup_property|Haagerup]] | yes | yes | yes | yes | yes | yes | | yes | yes | | | ^[[haagerup_property|Haagerup]] | yes | yes | yes | yes | yes | yes | | yes | yes | | | ||
- | ^[[amenability|Amenable]] | | $N=2$ | | | $N=4$ | yes | | | | | | + | ^[[amenability|Amenable]] | no | $N=2$ | | | $N=4$ | yes | | no | $N=2$ | | |
[(t:FO>Proven for $N\neq 3$)] | [(t:FO>Proven for $N\neq 3$)] |