Coarse median structures on groups


We study Bowditch’ concept of a coarse median on many examples of metric spaces and groups. A coarse median on a metric space is a ternary operation on the underlying set which satisfies the axioms of a median algebra up to bounded error (in a certain precise sense). We introduce the concept of a coarse median structure as an equivalence class of coarse medians up to uniformly bounded error. We define the pushforward and the pullback of a coarse median structure via a quasi-isometry. In terms of the pushforward/pullback notation, we note that the coarse median structure on a Gromov hyperbolic geodesic metric space is stable under quasi-isometries. Using the pullback construction, we describe a coarse median structure on spaces with measured walls which are at finite Hausdorff distance to their associated median spaces. We apply this to the measured wall structure on the real hyperbolic space and to the discrete wall structures obtained from Wise’s cubulation of classical small cancellation groups. We prove the existence of “approximating CAT(0) cubical complexes” in coarse median spaces and thereby generalize the fundamental result on approximating trees in Gromov hyperbolic geodesic spaces. We extend Bowditch’ theory of coarse median groups from finitely generated discrete groups to compactly generated locally compact groups. We show that a compactly generated locally compact coarse median group is compactly presented. Among finitely generated groups, we compare known permanence results concerning the classes of coarse median groups, a-T-menable groups, and groups with Kazhdan’s property (T). We present the explicit construction due to Ballmann-Swiatkowski of finitely generated hyperbolic groups with Kazhdan’s property (T). We show that there exist many examples of non-hyperbolic groups with Kazhdan’s property (T) which admit a left-equivariant coarse median.

Master’s thesis, University of Vienna