Amenable groups and Hadamard spaces with a totally disconnected isometry group

Abstract

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 ${\partial }_{\infty }^{\mathrm{fine}}X$ which is a refinement of the visual boundary ${\partial }_{\infty }X$. For each $x\in {\partial }_{\infty }^{\mathrm{fine}}X$, the stabilizer $G_x$ is amenable.