zbMATH — the first resource for mathematics

Amenability and Kunze-Stein property for groups acting on a tree. (English) Zbl 0671.43003
Let X be a locally finite tree and Aut(X) the locally compact group of all isometries of X. It is proved that a closed subgroup G of Aut(X) is amenable if and only if G has one of the following properties: (i) G fixes a vertex; (ii) G leaves invariant an edge; (iii) G fixes an end of X; (iv) G leaves invariant a pair of ends of X.
A locally compact group G is said to be a Kunze-Stein group if $$L^ p(G)*L^ 2(G)\subset L^ 2(G)$$ for every $$1<p<2$$. Let X be a homogeneous tree and G a subgroup of Aut(X) acting transitively on the vertices and on an open subset of the boundary of X. It is shown that G is either amenable or a Kunze-Stein group. The proofs depend on results on J. Tits [Essays on topology and related topics, 188-211 (1970; Zbl 0214.513)].
Reviewer: M.B.Bekka

MSC:
 43A07 Means on groups, semigroups, etc.; amenable groups 20B27 Infinite automorphism groups 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 43A70 Analysis on specific locally compact and other abelian groups 05C05 Trees
Full Text: