Quasi-isometric Classification of Graph Manifolds Groups

We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of graph-manifolds with boundary in terms of certain finite two-colored graphs. A corollary is a quasi-isometry classification of Artin groups whose presentation graphs are trees. In particular any two right-angled Artin groups whose presentation graphs are trees of diameter greater than 3 are quasi-isometric, answering a question of Bestvina; further, this quasi-isometry class does not include any other right-angled Artin groups. A finitely generated group can be considered geometrically when endowed with a word metric—up to quasi-isometric equivalence, such metrics are unique. (Henceforth only finitely generated groups will be considered.) Given a collection of groups, G, Gromov proposed the fundamental questions of identifying which groups are quasi-isometric to those in G (rigidity) and which groups in G are quasi-isometric to each other (classification) [9]. In this paper, we focus on the classification question for graph manifold groups and right-angled Artin group; a graph manifold is a 3-manifold that can be decomposed along embedded tori and Klein bottles into finitely many Seifert manifolds. The minimal such decomposition is called the geometric decomposition. In this paper graph manifolds will always be orientable and irreducible. Kapovich and Leeb [11] proved that any group quasi-isometric to the universal cover of a non-geometric Haken manifold with zero Euler characteristic is, up to finite groups, isomorphic to the fundamental group of such a manifold. In particular, this implies that the class of fundamental groups of graph manifolds is rigid. We answer the classification question for closed non-geometric graph manifolds, resolving a question of Kapovich and Leeb [12] Theorem 2.1. Any two closed non-geometric graph manifolds have bilipschitz homeomorphic universal covers. In particular, their fundamental groups are quasiisometric. This contrasts with commensurability of closed graph manifolds: already in the case that the graph manifold is composed of just two Seifert pieces there are infinitely many commensurability classes (they are classified in that case but not in general, see Neumann [14]). 2000 Mathematics Subject Classification. Primary 53C20; Secondary 20F67, 20F69.