The end compactification |Γ| of a locally finite graph Γis the union of the graph and its ends, endowed with a suitable topology. We show that π₁(|Γ|) 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. We also show how to recover the standard description of π₁(|Γ|) given by Diestel and Sprüssel (2011). Finally, we give some applications of our result, including a short proof that certain loops in |Γ| are non-nullhomologous.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-2,
author = {Isaac Goldbring and Alessandro Sisto},
title = {The fundamental group of a locally finite graph with ends-a hyperfinite approach},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {21-39},
zbl = {1331.05231},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-2}
}
Isaac Goldbring; Alessandro Sisto. The fundamental group of a locally finite graph with ends-a hyperfinite approach. Fundamenta Mathematicae, Tome 233 (2016) pp. 21-39. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-2/