Cusps and commensurability classes of hyperbolic 4–manifolds

Sell, Connor

doi:10.2140/agt.2023.23.3805

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Cusps and commensurability classes of hyperbolic $4$–manifolds

Connor Sell

Algebraic & Geometric Topology 23 (2023) 3805–3834

DOI: 10.2140/agt.2023.23.3805

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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Volume 23, issue 8 (2023)