Deformation spaces of Coxeter truncation polytopes

• Published in: Journal of the London Mathematical Society Vol. 106; no. 4; pp. 3822 - 3864
• Main Authors: Choi, Suhyoung, Lee, Gye‐Seon, Marquis, Ludovic
• Format: Journal Article
• Language: English
• Published: London Mathematical Society ; Wiley 01.12.2022
• Subjects: Geometric Topology; Group Theory; Mathematics; orbifold; discrete subgroup of Lie group; moduli space; 57M50; 57N16; 57S30 real projective structure; reflection group; Coxeter group; 20F55; 2010 Mathematics Subject Classification. 22E40
• ISSN: 0024-6107; 1469-7750
• DOI: 10.1112/jlms.12675

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$\Gamma$ of the group of projective transformations so that the Γ$\Gamma$‐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$\Omega$ of the Γ$\Gamma$‐orbit of P$P$ is convex and Γ$\Gamma$ acts properly discontinuously on Ω$\Omega$. A Coxeter polytope P$P$ is 2$\hskip.001pt 2$‐perfect if P∖Ω$P \smallsetminus \Omega$ 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.