Amenable groups and Hadamard spaces with a totally disconnected isometry group

Abstract

sup { vertical-align: 0.8ex; font-size:75%; } sub { vertical-align: -0.8ex; font-size:85%; } .ionss { line-height: 1.8; } .ionss sub {position: relative; top: 4; left: 1; } .ionss sup {position: absolute; top: 70; left:985; } 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 ∂fine∞  X which is a refinement of the visual boundary ∂∞ X. For each x ∈ ∂fine∞  X, the stabilizer Gx is amenable.