The trace field of 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.

Hyperbolic manifolds and pseudo-arithmeticity

“…The exact description of the trace field K -and in particular the degree of K/kdepends on the pieces M i and the way they are glued together. A precise treatment is the subject of a separate article by the second author [20]. See also [21].…”

“…[8,Sect. 6.2] and of the second author [20]) that Δ 5 is not commensurable with any lattice obtained by gluing arithmetic pieces. At this point it is natural to ask: Note that by definition the trace field K of a pseudo-arithmetic lattice is totally real.…”

Embedding arithmetic hyperbolic manifolds

“…Indeed, Theorem 1.1 applies to many of the Gromov-Piatetski-Shapiro non-arithmetic manifolds [15] (several explicit 2-and 3-dimensional examples can be constructed, c.f. [29] for a 4-dimensional one) and their generalisations [14,27,28,34], as well as to the ones introduced by Agol [1] and Belolipetsky-Thomson [7] (c.f. also [26]).…”