The fundamental group of a locally finite graph with ends—a hyperfinite approach
The end compactification $|\varGamma |$ of a locally finite graph $\varGamma $ is the union of the graph and its ends, endowed with a suitable topology. We show that $\pi _1(|\varGamma |)$ 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 $\pi _1(|\varGamma |)$ given by Diestel and Sprüssel (2011). Finally, we give some applications of our result, including a short proof that certain loops in $|\varGamma |$ are non-nullhomologous.