Fundamental groups as limits of discrete fundamental groups

Abstract

In this note we investigate to what extent the fundamental group of a metric space can be described as the inverse limit of its discrete fundamental groups. We show that some mild conditions suffice to imply the existence of an isomorphism and we provide a list of counterexamples to possible weakenings of these hypotheses.

Discrete fundamental groups of warped cones and expanders

“…In fact, for a fixed metric space the inverse limit lim ← − π 1,θ (X, d), x 0 for θ → 0 is generally non trivial and it is often isomorphic to the fundamental group π 1 (X, x 0 ). This is the subject of [Vig18a].…”

“…Remark 5.1. It would be notationally more correct to denote θ-discretisations by γ θ (as we do in [Vig18a]). Still, in most of what follows the parameter θ will be fixed and hence we decided to drop it from the notation for aesthetic reasons.…”

“…Let H i : [0, 1] 2 → X be the homotopy between γ i and γ ′ i . By compactness, it is easy to see (see [Vig18a] for details) that there exists an N ∈ N large enough so that the maps H i : [N ] 2 → X defined by…”

“…It is proved in [Vig18a] (and it is implicit in [BCW14]) that the θ-homotopy class of γ does not depend on the specific choice of the times t i . Moreover, if two continuous paths γ and γ ′ are (freely) homotopic then any two θ-discretisations γ and γ ′ are (freely) θ-homotopic.…”

“…This is a sort of continuous version of [BKLW01,Theorem 2.7]. The techniques we use here are also similar in spirit to what we use in the proof of the main result of [Vig18a], but we had to include a proof because there are some additional difficulties.…”