Narrow Results

Edelsbrunner et al. defined a framework of shape deformations with shapes bounded by skin manifold. We prove that the infinitely many synthesized shapes in the deformation sequence share finitely many common Voronoi complexes. Therefore, we propose a new algorithm to compute the common Voronoi complexes efficiently for the deformations, and use these common complexes to compute the synthesized shapes in real time. This makes generating, visualizing, and customizing shape deformations feasible.

We show that for every finite-volume hyperbolic $3$-manifold $M$ and every prime $p$ we have $dim{H}_1(M;F_p)<168.602\cdot \mathrm{\text{vol}}(M)$. There are slightly stronger estimates if $p=2$ or if $M$ is non-compact. This improves on a result proved by Agol, Leininger and Margalit, which gave the same inequality with a coefficient of $334.08$ in place of $168.602$. It also improves on the analogous result with a coefficient of about $260$, which could have been obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if $M$ is a finite-volume orientable hyperbolic $3$-manifold such that ${\pi }_1(M)$ is $2$-semifree, then $\mathrm{rank}{\pi }_1(M)<1+\lambda 0\cdot \mathrm{\text{vol}}M$, where $\lambda 0$ is a certain constant less than $167.79$.