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

Isaac Goldbring, and Alessandro Sisto. "The fundamental group of a locally finite graph with ends-a hyperfinite approach." Fundamenta Mathematicae 232.1 (2016): 21-39. <http://eudml.org/doc/283295>.

@article{IsaacGoldbring2016,
abstract = {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.},
author = {Isaac Goldbring, Alessandro Sisto},
journal = {Fundamenta Mathematicae},
keywords = {ends of graphs; Freudenthal compactification; topological cycle space},
language = {eng},
number = {1},
pages = {21-39},
title = {The fundamental group of a locally finite graph with ends-a hyperfinite approach},
url = {http://eudml.org/doc/283295},
volume = {232},
year = {2016},
}

TY - JOUR
AU - Isaac Goldbring
AU - Alessandro Sisto
TI - The fundamental group of a locally finite graph with ends-a hyperfinite approach
JO - Fundamenta Mathematicae
PY - 2016
VL - 232
IS - 1
SP - 21
EP - 39
AB - 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.
LA - eng
KW - ends of graphs; Freudenthal compactification; topological cycle space
UR - http://eudml.org/doc/283295
ER -