Interactions between Coarse Homotopy and Ends on Proper Geodesic Spaces

TL;DR

This paper explores how the concept of ends in coarse geometry behaves under coarse homotopy, establishing invariance, relationships with coarse path components, and conditions for injectivity in specific spaces like geometric trees.

Contribution

It demonstrates that the set of ends is a coarse homotopy invariant and investigates the relationship between ends and coarse path components, including conditions for when they coincide.

Findings

Ends are coarse homotopy invariants in proper geodesic spaces.

There is a natural surjection from coarse path components to ends.

In locally finite geometric trees, the surjection is an injection.

Abstract

We consider the coarse-geometric notion of ends in the context of coarse homotopy. We show that, when recontextualized as a functor from an appropriate coarse category of proper geodesic spaces, the set of ends $\mathcal{E}\text{nds}(-)$ is a coarse homotopy invariant. Further, we prove the existence of a natural surjection from the coarse path component functor $\pi_0^{\text{Crs}}(-)$ to $\mathcal{E}\text{nds}(-)$, and show that in general, this is not an injection (even when restricted to locally finite planar graphs). Finally, we begin to consider when this injection indeed exists by showing that this is the case for locally finite geometric trees, providing a number of useful preliminary lemmas on the behaviour of geodesics in this context.