I'm viewing Chapter 10 of GTM276 which focuses on some properties of topological groups including amenability and property (T). A footnote says they are almost exclusive. What does it mean? Does it mean that if a group has amenability then it will likely not have property (T)? I guess it true because we can see the difference on associated Cayley graphs: Amenability implies few connectivity while property (T) guarantees certain connectivity.
2026-03-29 17:27:06.1774805226
What is the relationship between amenability and property (T)?
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The proof that amenability + property (T) implies compactness is actually straightforward: If the locally compact group $G$ is amenable, then the left regular representation is faithful, hence it weakly contains the trivial representation ($g→1$). From property (T) it then follows that the regular representation truly contains the trivial representation. This means that there is some $f$ in $L^2(G)$ that is fixed under all left translations, hence $f$ must be constant. However if a constant function is square integrable, the underlying measure, namely Haar measure, must be finite. Finally, it is well known that Haar measure is only finite on compact groups.