| Résume | A well-known result from geometric group theory states that if a Cayley graph of a finitely generated group is quasi-isometric to a tree, then the group is virtually free. We prove an analogue of this result in the context of countable Borel equivalence relations, thereby answering a question posed by R. Tucker-Drob in 2015. Our theorem states more generally that if each connected component of a locally finite Borel graph 𝐺 is "tree-like" (e.g. is abstractly quasi-isometric to a tree or has bounded treewidth) then there exists an acyclic Borel graph with the same connected components as 𝐺. Our proof exploits the Stone duality between certain families of half-spaces in a graph and median graphs (the 1-skeleta of CAT(0) cube complexes), via clopen ultrafilters on these families of half-spaces. This is joint work with R. Chen, A. Poulin, and R. Tao. |
