The trace field of hyperbolic gluings

The first part of the thesis focuses on the study of the adjoint trace field of hyperbolic manifolds, an algebraic invariant introduced by Vinberg and generalizing the invariant trace field of dimension 3. We compute the adjoint trace field in the case of general hyperbolic gluings, and show how it can be used to prove that certain gluings are nonarithmetic. In the second part we introduce and motivate the concept of pseudo-arithmeticity, a notion that encompasses all known examples of hyperbolic manifolds. We then prove that gluings of arithmetic pieces are always pseudo-arithmetic, using results from the first part