Discrete fundamental groups of warped cones and expanders

Abstract

In this paper we compute the discrete fundamental groups of warped cones. As an immediate consequence, this allows us to show that there exist coarsely simply-connected expanders and superexpanders. This also provides a strong coarse invariant of warped cones and implies that many warped cones cannot be coarsely equivalent to any box space.

Super-expanders and warped cones

“…This is a consequence of an impressive dynamical quasiisometric-rigidity result for warped cones that they obtain. An important tool employed are coarse fundamental groups, which were also independently studied in a very interesting work [39] by Federico Vigolo.…”

“…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…”

Fundamental groups as limits of discrete fundamental groups

“…Under the assumption of some 'niceness' properties, this fact can be broadly generalised. For example, in [9] we show that if X is a geodesic metric space then π 1,θ (X) is isomorphic to the quotient of π 1 (X) where all the loops of length at most 4θ (up to free homotopy) are killed. In this sense we can say that the θ-fundamental group is the quotient of the fundamental group where short cycles are ignored.…”

“…The study of the groups π 1,θ (X, x 0 ) already proved to be very fruitful (see for example the survey [3]). When the parameter θ is large it can provide rather strong coarse invariants [4,6,9]. On the other hand, the behaviour of π 1,θ (X, x 0 ) as θ tends to zero has not yet been investigated thoroughly.…”

“…This note was born as a spin-off of [9]. In that paper, we utilise the relation between the fundamental group and π 1,θ (X, x 0 ) for large θ to produce coarse invariants of warped systems.…”