I was wondering if there are quasi-isometric (complete) metrics on the once-punctured torus and the Loch Ness Monster. By an once-punctured torus I mean the 1-genus surface with one puncture and by the Loch Ness Monster I mean the orientable surface of infinite genus with an unique end (see this question of mine for concrete definitions) which is necessarily non-planar.
For instance, the once-punctured torus admits a hyperbolic metric for which the puncture is identified with a hyperbolic cusp isometric to the product $S^1\times [1,+\infty)$, where $S^1$ is the circle equipped with a flat metric $dx$ and the metric on the product is given by the Riemannian metric $$\frac{dx^2+dt^2}{t^2}.$$
Thus, according to Sam Nead's answer on this post, the once-punctured torus is quasi-isometric to a ray.
On the other hand, thinking on the end space of the Loch Ness Monster in terms of proper rays (See my previously cited post for a concrete description), I would be tempted to suppose that the Loch Ness Monster carries a metric for which a geodesic ray escaping to its unique end makes the surface quasi-isometric to a ray as well, in analogy with the once-punctured torus and the geodesic rays escaping to its hyperbolic cusp.
In case that my question had a negative answer, at least can we find a metric in the infinitely non-orientable surface of infinite genus with an unique end (which is necessarily non-orientable) quasi-isometric to either of the two surfaces above under suitable (complete) metrics?.