approximation_properties_summary
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approximation_properties_summary [2019/10/04 10:09] d.gromada |
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) | ||
| + | |||
| + | Amenable ⇒ Haagerup | ||
| + | |||
| + | Property (T) and (F) ⇒ Residually finite | ||
| + | |||
| + | Property (T) and Haagerup ⇔ finite | ||
| ===== Stability results ===== | ===== Stability results ===== | ||
| - | | ::: ^ Products ^^^^ Subgroups ^ Quotients ^ Dual top. gen. ^ | + | | ^ Products ^^^^ Subgroups ^ Quotients ^ Dual top. gen. ^ |
| - | | ^ Direct ^ Free ^ Wreath ^ Free wr. ^ ::: ^ ::: ^ ::: ^ | + | | ::: ^ Direct ^ Free ^ Wreath ^ Free wr. ^ ::: ^ ::: ^ ::: ^ |
| - | ^[[kirchberg_property|Kirchberg (F)]] | | + | ^[[kirchberg_property|Kirchberg (F)]] | | | | | | | | |
| - | ^[[residual_finiteness|Resid. finite]]| | + | ^[[residual_finiteness|Resid. finite]]| | | | | | | yes | |
| - | ^[[kazhdan_property|Kazhdan (T)]] | | + | ^[[kazhdan_property|Kazhdan (T)]] | | | | | | yes | | |
| - | ^[[haagerup_property|Haagerup]] | | + | ^[[haagerup_property|Haagerup]] | | yes | | | yes | | | |
| - | ^[[amenability|Amenable]] | | + | ^[[amenability|Amenable]] | yes | no | | | yes | | | |
| ===== Concrete examples ===== | ===== Concrete examples ===== | ||
| - | | ^ free CMQG duals (assuming $N$ high enough) ^^^^^ | + | | ^ free CMQG duals (assuming $N$ high enough) ^^^^^ half-lib. duals ^^ non-unimod. ^^ Groups ^ |
| - | | ::: ^ $\hat U_N^+$ ^ $\hat O_N^+$ ^ $\hat H_N^+$ ^ $\hat H_N^{s+}$^$\hat S_N^+$^ | + | | ::: ^ $\hat U_N^+$ ^ $\hat O_N^+$ ^ $\hat H_N^+$ ^ $\hat H_N^{s+}$^$\hat S_N^+$^ $\hat O_N^*$ ^ $\hat H_N^*$ ^ $\hat U_F^+$ ^ $\hat O_F^+$ ^ ^ |
| - | ^[[kirchberg_property|Kirchberg (F)]] | yes [(t:FO)] | yes [(t:FO)] | yes[(t:FH)] | yes[(t:FH)] | yes | | + | ^[[kirchberg_property|Kirchberg (F)]] | yes [(t:FO)] | yes [(t:FO)] | yes[(t:FH)] | yes[(t:FH)] | yes | | | | | |
| - | ^[[residual_finiteness|Resid. finite]]| yes [(t:FO)] | yes [(t:FO)] | yes[(t:FH)] | yes[(t:FH)] | yes | | + | ^[[residual_finiteness|Resid. finite]]| yes [(t:FO)] | yes [(t:FO)] | yes[(t:FH)] | yes[(t:FH)] | yes | | | no || |
| - | ^[[kazhdan_property|Kazhdan (T)]] | no | no | no | no | no | | + | ^[[kazhdan_property|Kazhdan (T)]] | no | no | no | no | no | no | | no || |
| - | ^[[haagerup_property|Haagerup]] | yes | yes | yes | yes | yes | | + | ^[[haagerup_property|Haagerup]] | yes | yes | yes | yes | yes | yes | | yes | yes | | |
| - | ^[[amenability|Amenable]] | | $N=2$ | | | | | + | ^[[amenability|Amenable]] | no | $N=2$ | | | $N=4$ | yes | | no | $N=2$ | | |
| [(t:FO>Proven for $N\neq 3$)] | [(t:FO>Proven for $N\neq 3$)] | ||
| [(t:FH>Proven for $N\ge 4$)] | [(t:FH>Proven for $N\ge 4$)] | ||
| + | |||
| + | In addition, any finite quantum group satisfies all the listed approximation properties. Any Abelian discrete quantum group (dual of a compact group) satisfies all the listed approximation properties except for property (T) (Abelian discrete QG has (T) iff it is finite). | ||
| ~~REFNOTES t ~~ | ~~REFNOTES t ~~ | ||
approximation_properties_summary.1570183768.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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