Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold
Let X be a simply connected manifold of nonpositive curvature and X(\(\infty)\) be the points at infinity in the sense of Eberlein-O'Neill. In this paper we examine the structure of the group of isometries stabilizing a positive Borel measure on X(\(\infty)\). In this general setting we prove that this group either fixes a point of X(\(\infty)\) or leaves invariant a flat subspace of X. As a corollary we obtain the fact that if the fundamental group of a manifold of finite volume and nonpositive curvature is amenable then this manifold is flat [\textit{A. Avez}, C. R. Acad. Sci., Paris, Sér. A 270, 188-191 (1970; Zbl 0188.264) and \textit{R. J. Zimmer}, Am. J. Math. 105, 1011-1022 (1983; Zbl 0527.57015)]. We also obtain that any amenable subgroup of the fundamental group of a compact manifold of nonpositive curvature is a Bieberbach group, i.e. a finite extension of a finitely generated free abelian group (\textit{Anderson}, to appear in Math. Ann.). We also examine the situation when X is a symmetric space of noncompact type and prove that in this case the stabilizer of a positive Borel measure on X(\(\infty)\) is amenable.
scientific article; zbMATH DE number 3967356