Related papers: Commensurability of 1-cusped hyperbolic 3-manifold…

We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.

It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there…

This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.

In this paper we prove that if $M_K$ is the complement of a non-fibered twist knot $K$ in $\mathbb S^3$, then $M_K$ is not commensurable to a fibered knot complement in a $\mathbb Z/ 2 \mathbb Z$-homology sphere. To prove this result we…

We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at…

Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3-manifolds that share an arbitrarily large…

We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of…

This paper initiates a systematic study of the relation of commensurability of surface automorphisms, or equivalently, fibered commensurability of 3-manifolds fibering over the circle. We show that every hyperbolic fibered commensurability…

We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an…

We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable,…

We show that if the fundamental group of the complement of a rationally homologically fibered knot in a rational homology 3-sphere is bi-orderable, then its Alexander polynomial has at least one positive real root. Our argument can be…

This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the…

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In…

We introduce a class of cusped hyperbolic $3$-manifolds that we call mixed-platonic, composed of regular ideal hyperbolic polyhedra of more than one type, which includes certain previously-known examples. We establish basic facts about…

Let $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in $\mathbb S^3$. Then, if $M$ admits an exhaustion by $\pi_1$-injective sub-manifolds there exists cantor sets $C_n\subset \mathbb S^3$ such that $N_n=\mathbb…

This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding…

We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to…

We discuss fibered commensurability of fibrations on a hyperbolic 3-manifold, a notion introduced by Calegari, Sun and Wang. We construct manifolds with non-symmetric but commensurable fibrations on the same fibered face. We also prove that…

In this paper we provide the first examples of arithmetic hyperbolic 3-manifolds that are rational homology spheres and bound geometrically either compact or cusped hyperbolic 4-manifolds.