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Interactions between Coarse Homotopy and Ends on Proper Geodesic Spaces

Bradley Ashley

TL;DR

The paper investigates how coarse ends $\mathcal{E}\text{nds}(-)$ interact with coarse path components $\pi_0^{\text{Crs}}(-)$ for proper geodesic spaces. It shows ends form a coarse-homotopy invariant functor and constructs a natural surjection from $\pi_0^{\text{Crs}}(-)$ to $\mathcal{E}\text{nds}(-)$, which in general is not injective. By analyzing locally finite geometric trees, the authors identify a subcategory where the surjection is an isomorphism, proving $\mathcal{E}\text{nds}(-) \cong \pi_0^{\text{Crs}}(-)$ for locally finite geometric trees and for finitely generated virtually free groups. These results illuminate when coarse path components and ends coincide and point to refinements needed to capture ends in more complex spaces.

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 $π_0^{\text{Crs}}(-)$ 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.

Interactions between Coarse Homotopy and Ends on Proper Geodesic Spaces

TL;DR

The paper investigates how coarse ends interact with coarse path components for proper geodesic spaces. It shows ends form a coarse-homotopy invariant functor and constructs a natural surjection from to , which in general is not injective. By analyzing locally finite geometric trees, the authors identify a subcategory where the surjection is an isomorphism, proving for locally finite geometric trees and for finitely generated virtually free groups. These results illuminate when coarse path components and ends coincide and point to refinements needed to capture ends in more complex spaces.

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 is a coarse homotopy invariant. Further, we prove the existence of a natural surjection from the coarse path component functor to , 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.
Paper Structure (9 sections, 51 theorems, 100 equations)

This paper contains 9 sections, 51 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Large-Scale Geometry
  3. Ends
  4. Abstract Coarse Geometry
  5. Coarse Homotopy
  6. Coarse Path Components
  7. Coarse Homotopy and Ends of Proper Geodesic Spaces
  8. Geodesics in Locally Finite Geometric Trees
  9. Coarse Homotopy and Ends of Locally Finite Geometric Trees

Key Result

Proposition 2.12

Let $X$ be a geodesic metric space, and $x,x',x"$ be points in $X$ such that we have $x'\in \text{im}(u_{x,x"})$. Then the map $u_{x,x"}:[0,(u_{x,x"})^{-1}(x')] \to X$ is a geodesic from $x$ to $x'$, and the map $u_{x,x"}:[(u_{x,x"})^{-1}(x'),a_{x,x"}] \to X$ is a geodesic from $x'$ to $x"$.

Theorems & Definitions (133)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Example 2.10
  • Definition 2.11
  • ...and 123 more