Discrete fundamental groups of Warped Cones and expanders

TL;DR

This paper computes the discrete fundamental groups of warped cones, revealing new properties of expanders and superexpanders, and establishing coarse invariants that distinguish warped cones from box spaces.

Contribution

It introduces the computation of discrete fundamental groups of warped cones and demonstrates their implications for expanders, superexpanders, and coarse geometry.

Findings

Existence of coarsely simply-connected expanders and superexpanders.

Discrete fundamental groups serve as coarse invariants for warped cones.

Many warped cones are not coarsely equivalent to box spaces.

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.