The fundamental group of a locally finite graph with ends: a hyperfinite approach

TL;DR

This paper investigates the fundamental group of the end compactification of a locally finite graph, showing it embeds into a hyperfinite free group via nonstandard analysis, extending previous results.

Contribution

It introduces a hyperfinite approach to embed the fundamental group into an ultraproduct of free groups, providing a new perspective on the structure of such groups.

Findings

Embedding into a hyperfinite free group established

Constructs an inverse limit of free groups for the embedding

Recovers a known result of Diestel and Sprüssel

Abstract

The end compactification |\Gamma| of the locally finite graph \Gamma is the union of the graph and its ends, endowed with a suitable topology. We show that \pi_1(|\Gamma|) embeds into a nonstandard free group with hyperfinitely many generators, i.e. an ultraproduct of finitely generated free groups, and that the embedding we construct factors through an embedding into an inverse limit of free groups, recovering a result of Diestel and Spr\"ussel.