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Cusps and commensurability classes of hyperbolic $4$–manifolds

Connor Sell

Algebraic & Geometric Topology 23 (2023) 3805–3834
Abstract

There are six orientable compact flat 3–manifolds that can occur as cusp cross-sections of hyperbolic 4–manifolds. We provide 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 type.

Keywords
hyperbolic manifold, $4$–manifold, cusp, arithmetic, quadratic form, hyperbolic geometry
Mathematical Subject Classification
Primary: 57M50
Secondary: 11E20, 11F06, 16H05, 57K50
References
Publication
Received: 11 October 2021
Revised: 17 May 2022
Accepted: 8 June 2022
Published: 5 November 2023
Authors
Connor Sell
Department of Mathematics
Rice University
Houston, TX
United States

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