Arithmeticity and Hidden Symmetries of Fully Augmented Pretzel Link Complements

TL;DR

This paper investigates the arithmetic properties, symmetries, and invariant trace fields of fully augmented pretzel link complements, revealing new families with maximal hidden symmetries growth and explicit CM-field realizations.

Contribution

It precisely characterizes when these link complements are arithmetic, identifies commensurability classes, and constructs new non-arithmetic families with maximal hidden symmetries growth.

Findings

Identified exactly when link complements are arithmetic.

Constructed infinite families with no hidden symmetries and maximal hidden symmetries growth.

Explicitly computed invariant trace fields for these manifolds.

Abstract

This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite families of non-arithmetic fully augmented link complements: one that has no hidden symmetries and the other where the number of hidden symmetries grows linearly with volume. This second family realizes the maximal growth rate for the number of hidden symmetries relative to volume for non-arithmetic hyperbolic 3-manifolds. Our work requires a careful analysis of the geometry of these link complements, including their cusp shapes and totally geodesic surfaces inside of these manifolds.