On the end depth and ends of groups

TL;DR

This paper proves that finitely generated one-ended groups have linear end depth and offers alternative proofs connecting group growth to the number of ends.

Contribution

It establishes the linearity of end depth for one-ended groups and provides new proofs for theorems linking group growth and ends.

Findings

Finitely generated one-ended groups have linear end depth.

Alternative proofs for theorems relating group growth to the number of ends.

Abstract

We prove that any finitely generated one ended group has linear end depth. Moreover, we give alternative proofs to theorems relating the growth of a finitely generated group to the number of its ends.