# On the space of ends of infinitely generated groups

## TL;DR

This paper explores the topological structure of the space of ends in infinitely generated groups, revealing diverse behaviors and establishing new connections with concepts like the Stone-Cech compactification.

## Contribution

It characterizes the space of ends for various classes of infinitely generated groups, showing when they are metrizable or map onto the Stone-Cech compactification.

## Key findings

- For free product Z*Q, the space of ends is a Cantor space.
- For infinite rank free groups, the space of ends is non-metrizable.
- The space of ends is either metrizable or maps onto the Stone-Cech compactification of N.

## Abstract

We study the space of ends of groups. For a finitely generated group, this is a Cantor space as soon as it is infinite. In contrast, we show that for infinitely generated countable groups, it exhibits several behaviors. For instance, we show that for the free product Z*Q, it is a Cantor space, while for a free group of infinite rank, it is not metrizable. For arbitrary countable groups, we actually establish an alternative: the space of ends is either metrizable, or has a continuous map onto the Stone-Cech compactification of N. We also show that the space of ends of a countable group has a continuous map onto the Stone-Cech boundary of N if and only if the group is infinite locally finite, and that otherwise it is separable. For arbitrary groups, we also prove that the space of ends, if infinite, has no isolated point. We also consider these questions for locally compact groups; for instance we extend Holt's theorem by showing that non-sigma-compact regionally elliptic groups are 1-ended.