The trace field of hyperbolic gluings

TL;DR

This paper introduces a method to determine the adjoint trace field of hyperbolic manifold gluings, enabling the identification of nonarithmetic examples and advancing understanding of their geometric properties.

Contribution

It provides a new technique to prove nonarithmeticity of hyperbolic gluings, applicable to classical and generalized constructions, and offers new nonarithmetic examples.

Findings

Identified many new nonarithmetic gluings

Proved the noncommensurability of the Coxeter 5-simplex with arithmetic pieces

Developed a method to determine the adjoint trace field for hyperbolic gluings

Abstract

We determine the adjoint trace field of gluings of general hyperbolic manifolds. This provides a new method to prove the nonarithmeticity of gluings, which can be applied to the classical construction of Gromov and Piatetski-Shapiro (and generalizations) as well as certain gluings of pieces of commensurable arithmetic manifolds. As an application we give many new examples of nonarithmetic gluings and prove that the unique nonarithmetic Coxeter 5-simplex is not commensurable to any gluing of arithmetic pieces.