Regular simplices and lower volume bounds for hyperbolic \(n\)-manifolds
In this very clearly written paper the author essentially extends the lower volume bounds for hyperbolic \(n\)-manifolds due to \textit{R. Meyerhoff} for \(n=3\) [Comment. Math. Helv. 61, 271-278 (1986; Zbl 0611.57010)]. Let \(M\) be a hyperbolic \(n\)-manifold, and \(r(M)\) denote the in-radius of \(M\) associated to the largest ball embeddable in \(M\). Let \(S_{\text{reg}}(2\alpha)\subset H^n\) denote the regular hyperbolic simplex with dihedral angle \(2\alpha\) and edge length \(2r(M)\) related by \[ \frac{1}{n}<\cos(2\alpha)=\frac{ch(2r(M))}{1+(n-1)ch(2r(M))}< \frac{1}{n-1}, \] let \(s_{\text{reg}}(2\alpha)\) further denote the spherical vertex simplex of \(S_{\text{reg}}(2\alpha)\). Let \(\text{vol}_n\) denote the \(n\)-dimensional volume function, thus \[ \Omega_n:=\text{vol}_n(S^n)=\frac{2\pi^{\frac{n+1}{2}}}{\Gamma (\frac{n+1}{2})} \] is the volume of the standard \(n\)-sphere. Theorem: For \(n\geq 2\), let \(M\) be a compact hyperbolic \(n\)-manifold with inradius \(r(M)\). Then \[ \text{vol}_n(M)\geq \frac{\Omega_{n-1}} {n+1}\frac{\text{vol}_n(S_{\text{reg}}(2\alpha))}{\text{vol}_{n-1}(s_{\text{reg}}(2\alpha))}=\Omega_{n-1} \frac {\text{vol}_n(R(\alpha))}{\text{vol}_{n-1}(r(\alpha))}, \] where in the latter term the Coxeter-Dynkin-Schläfli-Vinberg diagram \[ \sum(R(\alpha)):\;\underset {0} {\circ}\overset {^{\scriptstyle{\alpha}}} {\diagrbar}\circ\diagrbar\circ\diagrbar\;\dots\;\diagrbar\circ\diagrbar\underset {n} \circ \] describes the characteristic orthoscheme \(R(\alpha)\) of \(S_{\text{reg}}(2\alpha)\) with edge length \(\ell(P_0P_1)=r(M)\) and vertex figure \(r(\alpha)\) (by discarding the \(n\)th node from the diagram). The proof relies upon the local density bounds for hyperbolic sphere packings due to \textit{K. Böröczky} senior [Acta Math. Acad. Sci. Hung. 32, 243-261 (1978; Zbl 0422.52011)]. The author also analyses the critical points of her estimate, e.g. the behavior of Böröczky upper density \(d_n(r)\), the bound for hyperbolic simplex volume, the bound for the in-radius of \(M\), where many problems remain still open. Finally, the author discusses and applies her results to study volume spectra of compact hyperbolic \(n\)-manifolds and their minima for \(n\geq 2\).
scientific article; zbMATH DE number 839171
scientific article