Rigidity of warped cones and coarse geometry of expanders

Abstract

We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions are finite covers of conjugate actions. As a consequence, we produce continuous families of non-quasi-isometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental groups.From the above expression, it is clear the large-scale geometry of these level sets is invariant under taking finite quotients or extensions of the action: Lemma 2.3. Let F ⊆ Γ be a finite normal subgroup, and write Γ := Γ/F and M := M/F . Write M t (resp. M t ) for the level set at time t of the cone over Γ M (resp. Γ M ). Then M t is (1, B)-quasi-isometric to M t , where B is the maximal word length in Γ of an element of F .Proof. Write π : M t → M t for the natural quotient map. It is easy to see that π is distance non-increasing, i.e. 1-Lipschitz. For the lower bound, let C be the maximal word length of an element of the finite group F . Further

Spectral gap and origami expanders

“…Consequently, the O-graphs associated with these warped cones are a source of expanders, which is in fact very rich, as exemplified by the results of [FNL19]. In particular, Corollary B follows immediately from Theorem A by applying Theorem 2.18.…”

“…The spectral gap may simply be a consequence of Property (T) of the subgroup, but it can also occur rather generically for free subgroups [BG08b,BG12,BS16]. For subgroup actions, the discrete fundamental group of a level of the warped cone is infinite, as it contains the acting group or a group that surjects onto it [Vig19a,FNL19]. Hence, our warped cones are also not coarsely equivalent to any warped cone over an action from this large class.…”

“…This problem is highly nontrivial. The strongest to date coarse rigidity results obtained for box spaces [DK20] and warped cones [Vig19a,FNL19] make use of discrete fundamental groups, which vanish for many origami expanders by Theorem D. Other standard coarse invariants, such as Property A or asymptotic dimension are useless for expanders. Moreover, since all origami expanders come from warped cones over actions of a fixed group on manifolds of a fixed dimension, we expect their "local" properties to be the same.…”

Super-expanders and warped cones

“…As noticed in [19], it follows from quasi-isometricrigidity literature that our super-expanders cannot be quasi-isometric to many super-expanders of Lafforgue and Liao. David Fisher, Thang Nguyen, and Wouter van Limbeek [12] constructed a continuum of actions Γ Y yielding pairwise non-quasi-isometric warped cones satisfying the assumptions of Theorem 1.1 for all Banach spaces of non-trivial type. This is a consequence of an impressive dynamical quasiisometric-rigidity result for warped cones that they obtain.…”

“…It follows from computations in [12,39] that, for Y with a finite fundamental group, the coarse fundamental group of O Γ Y is virtually isomorphic to Γ. If O Γ Y is quasi-isometric to a Margulis expander (Λ/Λ i ), then…”