Ends of Uncountable Locally Finite Groups: An Analytic and Coarse Geometric Approach

Abstract

It is a classical result of D. F. Holt that an uncountable, locally finite discrete group possesses exactly one end. While traditional approaches rely on cohomological methods and the combinatorial manipulation of almost-invariant subsets, we establish this result through coarse geometry via the topological structure of the Higson corona. By interpreting the boundaries of coarsely clopen sets in terms of subgroup invariance, we evaluate their associated characteristic functions on the large-scale geometry of the group. Utilizing a Skolem-hull extraction and a periodic coarse rerouting technique, we force a geometric rigidity that prohibits multiple ends.