Amenable groups and Hadamard spaces with a totally disconnected isometry group

Pierre-Emmanuel Caprace (1, 2)

1 ULB - Université libre de Bruxelles
2 University of Oxford

Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set $\bdfine X$ which is a refinement of the visual boundary $\bd X$. For each $x \in \bdfine X$, the stabilizer $G_x$ is amenable.

Mots clés: amenable group; $\CAT0$ space; totally disconnected group; locally finite group

Domaines: Théorie des groupes [math.GR]

Pierre-Emmanuel Caprace. Amenable groups and Hadamard spaces with a totally disconnected isometry group. Commentarii Mathematici Helvetici, 2009, 84, pp.437-455. ⟨10.4171/MCH/168⟩. ⟨hal-00146369⟩