Let $M$ be a closed hyperbolic 3-manifold and $H_{g}$ a genus g handlebody. Assume that $\pi: int(H_{g})\rightarrow M$ is a cover. Denote $N\subset H_{g}$ the convex core.
My question is: If the diameter of $\pi(\partial N)$ is finite in $M$, is the diameter of $\pi(N) $ bounded by the diameter of $\pi(\partial N)$?
Thank you!
Take a Schottky group $\Gamma$ with convex core $N\subset \mathbb{H}^3/\Gamma$ with $diam(N)$ large, but $diam(\partial N)$ bounded. Then by a theorem of Robert Brooks, there is a small deformation $\Gamma_\epsilon$ of $\Gamma$ so that $\Gamma_\epsilon$ has a cocompact extension. The construction is by extension by a reflection group such that the convex core $N_\epsilon$ embeds in the quotient. Then the diameter of the projection will be arbitrarily large but the boundary remains bounded.
The existence of such $\Gamma$ follows from the theory of Kleinian groups. For example, one may take a handlebody $H$ and a mapping class $f:\partial H\to \partial H$ which is sufficiently “generic” (its stable lamination has positive intersection number with any lamination in the Masur domain of $H$), then modifying the conformal structure on the boundary of a Schottky group hyperbolic metric on $H$ by $f^n$ gives a sequence of hyperbolic manifolds which converge to the infinite-cyclic cover of the mapping torus on one end, and hence the approximates have diameter of the convex core approacing infinity but bounded geometry. This can be deduced from the proof in this paper.
