Quotient topology on the set of commensurability classes of hyperbolic 3-manifolds

Abstract

We investigate relation between Dehn fillings and commensurability of hyperbolic 3-manifolds. The set consisting of the commensurability classes of hyperbolic 3-manifolds admits the quotient topology induced by the geometric topology. We show that this quotient space satisfies some separation axioms. Roughly speaking, this means that commensurablity classes are sparsely distributed in the space consisting of the hyperbolic 3-manifolds.