good bi-infinite path in one-ended groups

Question

Let $G$ be an infinite, finitely generated group with one end. Let $d$ be a word-metric on the Cayley graph of $G$ with respect to some finite symmetric generating set $S$.

My question is:

Does there exist a good bi-infinite path $\{g_i\}_{i\in\mathbb{Z}}$ in $G$?

Here $\{g_i\}_{i\in\mathbb{Z}}$ is called a good bi-infinite path if $d(g_i, g_{i+1})=1$ for all $i$ and $\lim_{|i-j|\to\infty}d(g_{i}, g_j)=\infty$.

It is clear this path exists if $G$ is not torsion, so we may assume $G$ is an infinite torsion group.

Answer 1

Yes. It follows from König's lemma that every infinite, finitely-generated group $G$ has a bi-infinite geoedsic path in its Cayley graph.

To prove this, observe first that the Cayley graph of $G$ contains finite geodesic segments of arbitrary length, since there exist elements of $G$ with arbitrarily large distance from the identity.

Now consider the collection of all geodesic segments in the Cayley graph of length $2n$ ($n\geq 0$) that are centered on the identity vertex. Note that such segments exist for arbitrarily large values of $n$, since we can translate any geodesic path of length $2n$ to be centered on the identity vertex.

The collection of all such segments forms a locally finite rooted tree under inclusion, where the root is the segment of length $0$ and the vertices at depth $n$ are the segments of length $2n$. This tree is infinite, so by König's lemma it has at least one infinite ray starting at the root. The vertices of the tree that lie on this ray are geodesic segments that form a chain under inclusion, and the ascending union of these segments a bi-infinite geodesic path that goes through the identity vertex.