Convex projective manifolds with a cusp of any non‐diagonalizable type

Abstract

Recent work of Ballas, Cooper, and Leitner identifies (n + 1) types of n-dimensional convex projective cusps, one of which is the standard hyperbolic cusp. Work of Ballas-Marquis and Ballas-Danciger-Lee [Ballas, Danciger and Lee, 'Convex projective structures on nonhyperbolic threemanifolds',Geom. Topol. 22 (2018) 1593-1646] give examples of these exotic (non-hyperbolic) type cusps in dimension 3. Here an extension of the techniques of Ballas-Marquis shows the existence of all cusp types in all dimensions, except diagonalizable (type n) [Ballas, Cooper and Leitner, 'Generalized cusps in real projective manifolds: classification', Preprint, 2017, arXiv:1710.03132; Ballas and Marquis, 'Properly convex bending of hyperbolic manifolds', Preprint, 2016, arXiv:1609.03046].

Generalized cusps in real projective manifolds: classification

“…Thus, has a manifold compactification by adjoining . Recently, Martin Bobb produced the first examples of hyperbolic 3‐manifolds with type 2 cusps [6]. His construction works in arbitrary dimension to produce examples with different types of generalized cusp ends, but we only describe the 3‐dimensional version of his work here.…”

Codimension-$1$ Simplices in Divisible Convex Domains

“…There are some results on deforming cusped hyperbolic manifolds to change the cuspidal holonomy [DB19, BM16,BDL18]. The main result of [DB19] is that there is a manifold in every dimension with a cusp which can be of any type up to (d − 1), but not type d.…”

Constructing convex projective 3‐manifolds with generalized cusps

“…Until very recently, there were no known examples of a hyperbolic 3‐manifold with type 2 cusps. However, the author was recently made aware of work of M. Bobb [9] in which he produces the first examples of hyperbolic 3‐manifolds with a cusp of type 2. His methods are quite different from those of this paper and involve simultaneously bending along multiple embedded totally geodesic hypersurfaces.…”

“…Before proceeding we mention that there are other recent examples of manifolds admitting type 2 cusps due to Martin Bobb [9]; however, his examples involve a version of bending for arithmetic manifolds. By work of Reid [25], the figure‐eight knot is the only arithmetic knot complement, and so we see that the examples covered in this section are non‐arithmetic and hence not covered by Bobb's work.…”