We give a sufficient condition on the hyperplanes used in the inbreeding construction of Belolipetsky-Thomson to obtain nonarithmetic manifolds. We construct explicitly infinitely many examples of such manifolds that are pairwise non-commensurable and estimate their volume.Let M be a finite-volume hyperbolic n-manifold, with n ≥ 2. If M is complete, it can be written as a quotient Γ\H n for Γ a torsion-free lattice in the semi-simple Lie group PO(n, 1) ∼ = Isom(H n ), the group of isometries of the hyperbolic n-space H n . The lattice Γ is uniform if and only if M is compact.A standard way to construct hyperbolic manifolds in higher dimensions is via arithmetic lattices. In Lie groups with rank at least 2, this is actually the only possible construction (by Margulis' Arithmeticity Theorem); yet it is known that there are nonarithmetic lattices in PO(n, 1) for every n ≥ 2. Many examples in low dimensions were constructed by Vinberg using Coxeter groups (see [Vin67] and the references therein), but the first construction in arbitrary dimension was given by Gromov and Piatetski-Shapiro [GPS88]. Roughly, their idea consists in constructing two pieces of non-commensurable arithmetic manifolds with isometric boundaries and glueing them together to form a nonarithmetic manifold. This construction has then been generalized by Raimbault [Rai13] and Gelander and Levit [GL14] to produce many different commensurability classes of nonarithmetic manifolds.A similar construction was introduced by Belolipetsky and Thomson [BT11] to obtain manifolds with short systole. They start with two hyperplanes chosen at distance δ > 0 and find a torsion-free arithmetic lattice Γ such that, in M = Γ\H n , the hyperplanes project down to two disjoint hypersurfaces. Then they cut M open along the hypersurfaces and glue it back to a copy of itself along its boundary; as δ → 0, the systole of such a manifold then becomes arbitrarily small. Manifolds obtained via this construction will be referred to as doubly-cut glueings and the two corresponding hyperplanes as the cut hyperplanes (see Section 1.2).An interesting consequence is that infinitely many of such doubly-cut glueings are nonarithmetic and pairwise non-commensurable. Moreover, these are the first examples in arbitrary dimension of nonarithmetic manifolds that are quasi-arithmetic (see [Tho16]). However if one is only interested in constructing nonarithmetic manifolds, their proof is somehow nonexplicit in the sense that it relies on the systole argument for proving both nonarithmeticity and pairwise non-commensurability. Furthermore, it is hard to give an estimate on the volume of one particular nonarithmetic manifold.In this paper we give a sufficient condition on the cut hyperplanes to obtain nonarithmetic doubly-cut glueings. Recall that the group PO(n, 1) ∼ = Isom(H n ) has a natural matrix representation in O f (R) ⊂ GL n+1 (R) for f = −x 2 0 + x 2 1 + · · · + x 2 n the standard Lorentzian quadratic form (see Section 1.1).

We introduce and motivate a notion of pseudo-arithmeticity, which possibly applies to all lattices in PO ⁡ ( n , 1 ) \operatorname {PO}(n,1) with n > 3 n>3 . We further show that under an additional assumption (satisfied in all known cases), the covolumes of these lattices correspond to rational linear combinations of special values of L L -functions.

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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.

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.