Deformation spaces of Coxeter truncation polytopes - Choi, Suhyoung; Lee, Gye‐Seon; Marquis, Ludovic (Journal of the London Mathematical Society, 2022)
A convex polytope P$P$ in the real projective space with reflections in the facets of P$P$ is a Coxeter polytope if the reflections generate a subgroup Γ$Γ$ of the group of projective transformations so that the Γ$Γ$‐translates of the interior of P$P$ are mutually disjoint. It follows from work of Vinberg that if P$P$ is a Coxeter polytope, then the interior Ω$Ω$ of the Γ$Γ$‐orbit of P$P$ is convex and Γ$Γ$ acts properly discontinuously on Ω$Ω$. A Coxeter polytope P$P$ is 2$\hskip.001pt 2$‐perfect if P∖Ω$P \smallsetminus Ω$ consists of only some vertices of P$P$. In this paper, we describe the deformation spaces of 2$\hskip.001pt 2$‐perfect Coxeter polytopes P$P$ of dimensions d⩾4$d \geqslant 4$ with the same dihedral angles when the underlying polytope of P$P$ is a truncation polytope, that is, a polytope obtained from a simplex by successively truncating vertices. The deformation spaces of Coxeter truncation polytopes of dimensions d=2$d = 2$ and d=3$d = 3$ were studied, respectively, by Goldman and the third author.