arxiv:2109.12172

Cusps and Commensurability Classes of Hyperbolic 4-Manifolds

Published on Sep 24, 2021

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Abstract

The paper establishes criteria for the presence of specific cusp types in arithmetic hyperbolic 4-manifolds and provides examples of commensurability classes that lack certain cusp types.

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There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections 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 type.

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