Suppose $\Gamma$ is the fundamental group of a closed, oriented surface $S$. Let $B$ be a finitely generated, infinite index subgroup of $\Gamma$, and let $\Gamma_B$ be the compact core of the $B$-covering space of $S$ (well-defined up to isotopy in the $B$-covering space). Lifting hyperbolic structures gives a well-defined continuous function from $T(\Gamma)$, the Teichmuller space of hyperbolic structures $\Gamma$, to $T(\Gamma_B)$, the Teichmuller space of hyperbolic structures with totally geodesic boundary on $\Gamma_B$.

The general question I would like to ask is: How rigid is the map $T(\Gamma) \to T(\Gamma_B)$? However, as it stands that question suffers from the problem that the boundary components of $T(\Gamma_B)$ can be arbitrarily long. So, in order to sweep that problem under the rug, I want to compose with the map from $T(\Gamma_B)$ to the Masur-Minsky marking complex $M(\Gamma_B)$, defined in their paper "Geometry of the curve complex II".

Question 1: How rigid is the composed map $T(\Gamma) \to T(\Gamma_B) \to M(\Gamma_B)$?

EDIT: What I mean by "rigidity" here is, loosely speaking, some kind of measurement of the extent of the image of this map in the marking complex $M(\Gamma_B)$. Here are two examples which exhibit opposite extremes of rigidity.

For example, if $B$ is the fundamental group of an essential subsurface of $\Gamma$ then the composed map is essentially surjective. This is the "least rigid" possibility.

For another example, if $B$ has rank $2$ and $\Gamma_B$ is a pair of pants then the composed map has bounded image in $M(\Gamma_B)$. This is the "most rigid" possibility (although this example is uninteresting because $M(\Gamma_B)$ is itself just a single point).

Question 2: Are there examples of subgroups $B$ where $\Gamma_B$ is not just a pair of pants but the image of the composed map $T(\Gamma) \to M(\Gamma_B)$ has finite diameter?

As a reminder of how the map $T(\Gamma_B) \to M(\Gamma_B)$ is defined, one fixes an appropriate Margulis constant $\epsilon$ for $\Gamma_B$, chosen so that for any hyperbolic structure in $T(\Gamma_B)$ the collection of closed geodesics and closed proper arcs of length $\le \epsilon$ fill $\Gamma_B$, and from that collection one uses surgery to construct a pants decomposition of $\Gamma_B$ and a transverse closed curve for each pants curve.

This question occurred to me while pondering a question of Benjamin Steinberg. I am not aware of any literature on this question.

It would also make sense to ask this question with regard to a natural "lifting" map $M(\Gamma) \to M(\Gamma_B)$, but the image of this map is probably coarsely the same as the image of the map $T(\Gamma) \to M(\Gamma_B)$.

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