Commensurability and locally free Kleinian groups
نویسنده
چکیده
There are three main observations which make up the proof. The first observation, discussed in Section 2, is that convex co-compact subgroups of a fundamental group of a hyperbolic 3-manifold N persist in the approximates given by hyperbolic Dehn surgery. This result is stated formally as Proposition 2.1. The second observation, discussed in Section 4, is that a collection of distinct hyperbolic 2or 3-manifolds of uniformly bounded volume contains infinitely many commensurability classes. This result is stated formally as Proposition 4.1. Such collections of hyperbolic 3-manifolds are generated by hyperbolic Dehn surgery. The third observation, discussed in the proof of Theorem 5.1, is a general construction of finite volume hyperbolic 3-manifolds which is a direct application of Thurston’s hyperbolization theorem for Haken atoroidal 3-manifolds. The material in this Section is well-known, though to my knowledge the construction given has never been put to paper and so is included for the sake of completeness.
منابع مشابه
Incommensurability Criteria for Kleinian Groups
The purpose of this note is to present a criterion for an infinite collection of distinct hyperbolic 3-manifolds to be commensurably infinite. (Here, a collection of hyperbolic 3-manifolds is commensurably infinite if it contains representatives from infinitely many commensurability classes.) Namely, such a collection M is commensurably infinite if there is a uniform upper bound on the volumes ...
متن کاملExistence and Non-Existence of Torsion in Maximal Arithmetic Fuchsian Groups
In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Ei...
متن کاملThe space of Kleinian punctured torus groups is not locally connected
We show that the space of Kleinian punctured torus groups is not locally connected.
متن کاملInvariant Trace-fields and Quaternion Algebras of Polyhedral Groups
Let P be a polyhedron in H$ of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of IsomH$. Let Γ+(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ+(P) is a Kleinian group of finite covolume. Γ+(P) is called a polyhedral group. As discussed in [12] and [13] for example (see §2 below), asso...
متن کامل