Kirby Problem 3.23

Definitions and mathematical statement

Write an orientable complete finite-volume hyperbolic three-manifold as $M=\mathbb H^3/\Gamma$, with $\Gamma$ torsion-free and discrete in $\operatorname{Isom}^+(\mathbb H^3)$. A number $\varepsilon>0$ is a Margulis number for $M$ if, for every $x\in\mathbb H^3$ and $a,b\in\Gamma$, \[ \max\{d(x,ax),d(x,bx)\}<\varepsilon\quad\Longrightarrow\quad ab=ba. \] Let $\mu(M)$ be the supremum of such $\varepsilon$; equivalently it is the infimum of the displayed maximum over $x$ and noncommuting pairs $a,b$. Use displacement lengths, not half-lengths or injectivity radii, throughout.

Define the three-dimensional constant in the class specified by the source by \[ \mu_3=\inf_M\mu(M), \] where $M$ ranges over all orientable complete finite-volume hyperbolic three-manifolds. Determine $\mu_3$. More specifically, is it attained uniquely, up to isometry, by the Weeks manifold $W$, with the source's candidate value $\mu(W)\approx0.77442$? Here $W$ denotes the closed orientable manifold known as the Weeks manifold. A numerical value alone does not establish the universal lower bound or uniqueness, and orbifolds with torsion are not included in this infimum.

Short English statement

Find the optimal universal Margulis displacement bound for orientable finite-volume hyperbolic three-manifolds, and decide whether only the Weeks manifold attains it.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2821 (KP-3.23), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.23. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.