The end compactification Λ of a locally finite graph Λ is the union of the graph and its ends, endowed with a suitable topology. We show that π1(Λ) 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(Λ) given by Diestel and Sprüssel (2011). Finally, we give some applications of our result, including a short proof that certain loops in Λ are non-nullhomologous.
- End compactifiction of a locally finite graph
- Fundamental group
- Nonstandard analysis
ASJC Scopus subject areas
- Algebra and Number Theory