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

## TL;DR

This paper studies the topological structure of the space of hyperbolic 3-manifold classes under commensurability, revealing that these classes are sparsely distributed within the space.

## Contribution

It introduces a quotient topology on the set of commensurability classes and proves separation properties, advancing understanding of their distribution.

## Key findings

- The quotient space satisfies certain separation axioms.
- Commensurability classes are sparsely distributed.
- The relation between Dehn fillings and commensurability is clarified.

## 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.