9 problems

Let $G$ be a Lie group, let $H$ be a closed subgroup, and let $d(H)$ denote the invariant used in the paper to measure the dimension relevant to tessellations of $G/H$. One-dimensi…

Nonexistence conjecture. None of these homogeneous spaces has a tessellation. These are presented as three special cases of Kobayashi's general conjecture; the supplied source give…

Let $\Gamma$ be a prism of a hexagonal tessellation of the Klein bottle, and suppose its cutout contains an $m\times n$ array of tiles, with $m$ and $n$ even. Assume that both type…

Strict tessellation conjecture. $\Omega$ has a complete set of trigonometric eigenfunctions for the Laplace eigenvalue equation with the Dirichlet boundary condition if and only if…

Let $\sigma_1$ and $\sigma_2$ be valence sequences with $\sigma_1<\sigma_2$, and let $\underline{\lambda}_\sigma$ denote the lower growth-rate bound among face-homogeneous tessella…

Let $\sigma_1$ and $\sigma_2$ be valence sequences with $\sigma_1<\sigma_2$, and let $\underline{\lambda}_\sigma$ and $\overline{\lambda}_\sigma$ denote the lower and upper bounds…

Extremal growth-rate conjecture. The sets $\mathscr{L}_\sigma$ and $\mathscr{H}_\sigma$ are nonempty. Thus the lower and upper growth-rate bounds are realized by tessellations with…

Let $\sigma$ be the valence sequence of a face-homogeneous tessellation. Suppose that $\sigma$ contains $[p,q,r]$ as a subsequence, where $q$ is odd and $p\neq r$. Odd-valence subs…

Evenness conjecture. Every tetrahedral link complement has an even number of tetrahedra; equivalently, a corresponding combinatorial tetrahedral tessellation has an even number of…