Weakly amenable groups and amalgamated products. (English) Zbl 0780.43002
Let \(\{G_ i:i\in I\}\) be a family of amenable locally compact groups, each containing the compact open subgroup \(A\). Let \(G\) be the free product of the \(G_ i\) with amalgamation over \(A\): \(G=*_ AG_ i\). Then \(G\) is weakly amenable and \(\Gamma_ G = 1\) [definition and notation as in M. Cowling and U. Haagerup, Invent. Math. 96, 507-549 (1989; Zbl 0681.43012)]. As an example, \(G\) may be \(SL_ 2(\mathbb{Q}_ p)\) (though this example was already known, using the methods of J. de Cannière and U. Haagerup [Am. J. Math. 107, 455- 500 (1985; Zbl 0577.43002)].
Reviewer: M.Cowling (Kensington)
MSC:
| 43A35 | Positive definite functions on groups, semigroups, etc. |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
| 43A70 | Analysis on specific locally compact and other abelian groups |
| 43A77 | Harmonic analysis on general compact groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
