Corpus ID: 237940237

Cusps and Commensurability Classes of Hyperbolic 4-Manifolds

  title={Cusps and Commensurability Classes of Hyperbolic 4-Manifolds},
  author={Connor Sell},
There are six orientable, compact, flat 3-manifolds that can occur as cusp crosssections of hyperbolic 4-manifolds. This paper provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp… Expand

Tables from this paper


Hyperbolic 4-manifolds, colourings and mutations
We develop a way of seeing a complete orientable hyperbolic 4-manifold M as an orbifold cover of a Coxeter polytope P ⊂ H 4 that has a facet colouring. We also develop a way of finding a totallyExpand
All flat three-manifolds appear as cusps of hyperbolic four-manifolds
Abstract There are ten diffeomorphism classes of compact, flat 3-manifolds. It has been conjectured that each of these occurs as the boundary of a 4-manifold whose interior admits a complete,Expand
Peripheral separability and cusps of arithmetic hyperbolic orbifolds.
For X = R, C, or H, it is well known that cusp cross-sections of finite volume X-hyperbolic (n + 1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n +1)-dimensionalExpand
Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces
In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and applyExpand
Hyperbolic four-manifolds with one cusp
We introduce an algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologicallyExpand
Salem numbers and arithmetic hyperbolic groups
In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic formExpand
Arithmetic of hyperbolic 3-manifolds
This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the "arithmetic topology". Given 3-dimensional manifold M fibering over the circle we introduce an realExpand
All flat manifolds are cusps of hyperbolic orbifolds
We show that all closed flat n-manifolds are dieomorphic to a cusp crosssection in a nite volume hyperbolic n + 1-orbifold. AMS Classication 57M50; 57R99
Collisions at infinity in hyperbolic manifolds
Abstract For a complete, finite volume real hyperbolic n-manifold M, we investigate the map between homology of the cusps of M and the homology of M. Our main result provides a proof of a resultExpand
On the geometric boundaries of hyperbolic 4{manifolds
We provide, for hyperbolic and flat 3{manifolds, obstructions to bounding hyperbolic 4{manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.