When I say manifold below, I mean a complete orientable finite-volume hyperbolic $3$-manifold and when I say subsurface, I mean immersed closed totally geodesic subsurface.
Whenever a manifold has a subsurface, it has infinitely many different non-isometric ones. There were some nice articles a couple of years ago asking whether these can be used to determine the commensurability class of the manifold. In particular, you can organize this information as a genus spectrum and ask whether these being the same for two manifolds implies commensurability. (The answer is yes if the genus spectrum is nonempty and the manifolds are arithmetic but the answer is no in general.) As usual, I want to do something weird with this.
For a manifold $X$, let $\mathcal{S}_X(g,n)$ be the number of non-isometric surfaces in $X$ with genus $g$ and $n$ punctures. It has been shown that this is always finite. Hence as long as one of them is nonzero, there are infinitely many that are nonzero.
My question: when can we say that there infinitely many choices of $g,n$ so that $\mathcal{S}_X(g,n)\geq2$?
For instance, suppose we require that the manifold have two non-isometric subsurfaces with the same genus and number of punctures. Does this guarantee infinitely many other pairs of non-isometric surfaces, each pair sharing a distinct genus and number of punctures?
This article by Agol seems to suggest that there are always infinitely many pairs of the same genus, since it assumes that you can glue together infinitely many pairs of subsurfaces within the same manifold. He only writes out the details for dimension $4$, but says at the end that it applies for everything up to dimension $8$.
