Arithmetic of Hyperbolic Manifolds

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“…The invariant trace field k(M ) = k(Γ) is the field generated over Q by squares of traces of elements of Γ. It is the smallest field among trace fields of finite index subgroups of Γ ( [19], see also [13]). It is a number field and comes with a specific embedding in C.…”

“…In the non-compact case one can find genuine (rather than just degree one) ideal triangulations of M (see [7]) and the simplex parameters z i then lie in the invariant trace field k(M ) (see [13]). Thus β(M ) is the image of a class…”

“…e.g., [13]). It then either satisfies k = k and is CM-embedded or it satisfies k∩k ⊂ R. Thus, in the arithmetic case Theorem 1.4 and Conjecture 1.5 would say that rationality or irrationality of the Chern-Simons invariant is completely determined by whether or not k = k.…”

On commensurable hyperbolic Coxeter groups

“…Both, the field kG and the algebra AG are commensurability invariants (see [27]). Furthermore, kG is a finite non-real extension of Q (see [23,Theorem3.3.7]), and if the group G is not cocompact (containing parabolic elements), then the algebra AG is isomorphic to the matrix algebra M 2 (kG) (see [23,Theorem 3.3.8]).…”

On groups generated by two positive multi-twists: Teichmueller curves and Lehmer’s number

“…Given that the dilatations we are obtaining are naturally occurring as spectral radii of hyperbolic elements in certain non-elementary Fuchsian groups, we would be remiss not to mention the following (see [46], [37], and also [24]). Theorem 9.7 (Neumann-Reid) The Salem numbers are precisely the spectral radii of hyperbolic elements of arithmetic Fuchsian groups derived from quaternion algebras.…”