Property (TTT) and the Cowling-Haagerup constant (Q6622545)

Despite being mutually exclusive, amenability and Property (T) are very important and fruitful features that a locally compact group can possess. In his article, the author explores whether other properties of countable groups can be discovered by relaxing these two properties. Specifically, the author studies a weakening of the obstruction: if \(G\) is countable group and \(H\) a finite subgroup of \(G\) such that \((G,H)\) has relative Property (T), then \(G\) cannot be amenable. One of his key results is that by weakening relative Property (T), we obtain a weakened form of amenability: if \(G\) is countable group and \(H\) an finite subgroup of \(G\) such that \((G,H)\) has relative Property (TTT), then \(G\) becomes `weakly amenable' in the sense that its corresponding weak Haagerup constant is strictly greater than \(1\).