Amenable groups and Hadamard spaces with a totally disconnected isometry group

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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 X which is a refinement of the visual boundary X. For each x ∈ X, the stabilizer Gₓ is amenable.