PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 2, Pages 347–351 S 0002-9939(99)05027-3 Article electronically published on June 21, 1999 MINIMALLY ALMOST PERIODIC TOTALLY DISCONNECTED GROUPS CLAUDIO NEBBIA (Communicated by Roe Goodman) Abstract. In this paper we prove that every closed noncompact group G of isometries of a homogeneous tree which acts transitively on the tree boundary contains a normal closed cocompact subgroup G ′ which is minimally almost periodic. Moreover we prove that G ′ is a topologically simple group. 1. Introduction Let X be a homogeneous tree of finite order q +1 ≥ 3. We denote by Aut(X ) the locally compact group of all isometries of X with respect to the natural distance of X (d(x, y) is the length of the unique geodesic connecting x to y). We refer the reader to [2] for undefined notions and terminology. We fix x 0 ∈ X ; then the sets X + = {x ∈ X : d(x, x 0 ) is even} and X − = {x ∈ X : d(x, x 0 ) is odd} are the equivalence classes of the relation “d(x, y) is an even number”. Therefore this partition of X into the sets X + and X − is independent of the choice of x 0 . If G is a closed noncompact subgroup of Aut(X ) acting transitively on the tree boundary Ω, then either G acts transitively on X or G has exactly the orbits X + and X − [4, Prop. 2, pg. 143]. In particular if G has two orbits X + and X − , then every closed noncompact subgroup of G acting transitively on Ω has the same orbits of G. A notable example of this type is the subgroup Aut + (X ) of Aut(X ) generated by all rotations of X . More generally, let G be a closed subgroup of Aut(X ) acting transitively on X and Ω. Then the subgroup G + generated by all rotations of G is an open normal subgroup of G of index 2 acting transitively on Ω and having two orbits (X + and X − ) on X . In [8] J. Tits has proved that Aut + (X ) is an algebraically simple group. Furthermore, J. Tits proved that the group G + is algebraically simple for a larger class of groups with property (P) (see [8, 4.2, pg. 197]). Let G be a locally compact group; then G is said to be minimally almost periodic (briefly: m.a.p.) if every finite-dimensional unitary representation is trivial. This is equivalent to the fact that there is no continuous almost periodic function except constant functions. In the present paper we consider the class G of all closed subgroups G of Aut(X ) with the following properties: G acts transitively on Ω and G has two orbits on X . We prove that every group G ∈G contains one and only one nontrivial normal closed subgroup G ′ ∈G which is m.a.p., cocompact and topologically simple. This Received by the editors November 20, 1997 and, in revised form, March 31, 1998. 1991 Mathematics Subject Classification. Primary 20E08; Secondary 22D05, 43A60. c 1999 American Mathematical Society 347 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use