Injectivity radius from the number of generators

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $N=\mathbb H^3/\Gamma$ be a complete hyperbolic three-manifold. The limit set $\Lambda(\Gamma)$ consists of accumulation points at infinity of an orbit. Its convex core is the quotient by $\Gamma$ of the hyperbolic convex hull of $\Lambda(\Gamma)$. An embedded ball of radius $R$ is a metric ball isometric to a radius-$R$ ball of $\mathbb H^3$.

Statement recorded in the supplied source.

If $\pi_1(N)$ can be generated by $n$ elements, does there exist an upper bound $R_n$, depending only on $n$, for the radius of every embedded ball entirely contained in the convex core of $N$? [S2]

Short English statement

Does the number of generators alone bound above the sizes of all embedded balls lying entirely in a hyperbolic three-manifold’s convex core?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300088 (AMR-052-0088). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld (editor) and conference contributors, Conformal Dynamics Problem List, arXiv:math/9201271v1 (1990), Curt McMullen, §5, Convex hyperbolic 3-manifolds, Question, printed p. 12. https://arxiv.org/abs/math/9201271v1.