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

Isaac Goldbring ¹   ; Alessandro Sisto ²

¹ Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Science and Engineering Offices (M/C 249) 851 S. Morgan St. Chicago, IL 60607-7045, U.S.A.
² Department of Mathematics ETH Zürich HG J 14.4 Rämistrasse 101 8092 Zürich, Switzerland

Fundamenta Mathematicae, Tome 232 (2016) no. 1, pp. 21-39

Résumé

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.

Détail

DOI : 10.4064/fm232-1-2

Keywords: end compactification vargamma locally finite graph vargamma union graph its ends endowed suitable topology vargamma embeds nonstandard group hyperfinitely many generators ultraproduct finitely generated groups embedding construct factors through embedding inverse limit groups recover standard description vargamma given diestel spr ussel finally applications result including short proof certain loops vargamma non nullhomologous