Mutations and short geodesics in hyperbolic 3-manifolds

Millichap, Christian

doi:10.4310/cag.2017.v25.n3.a5

Cited by 18 publications

(21 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We make several remarks. (i)There are many other interesting work related the complex length with the geometry of hyperbolic three‐manifolds, see for instance . (ii)One may ask Question for the existence of

C^{0}

foliations. Our techniques rely on a Theorem of Sullivan which requires the taut foliation to be at least

C^{2}

. (iii)It is still unknown whether there exists any fibered hyperbolic three‐manifold which does admit a minimal foliation (

C^{0}

C^{2}

).…”

Section: Applicationsmentioning

confidence: 99%

Complex length of short curves and Minimal Fibrations of hyperbolic three‐Manifolds fibering over the circle

Huang

Wang

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

We investigate the maximal solid tubes around short simple closed geodesics in hyperbolic three‐manifolds and how the complex length of curves relates to closed least area incompressible minimal surfaces. As applications, we prove the existence of closed hyperbolic three‐manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces isotopic to the fiber. We also show the existence of quasi‐Fuchsian manifolds containing arbitrarily many embedded closed incompressible minimal surfaces. Our strategy is to prove main theorems under natural geometric conditions on the complex length of closed curves on a fibered hyperbolic three‐manifold, then by computer programs, we find explicit examples where these conditions are satisfied.

show abstract

C^{0}

foliations. Our techniques rely on a Theorem of Sullivan which requires the taut foliation to be at least

C^{2}

. (iii)It is still unknown whether there exists any fibered hyperbolic three‐manifold which does admit a minimal foliation (

C^{0}

C^{2}

).…”

Section: Applicationsmentioning

confidence: 99%

Complex length of short curves and Minimal Fibrations of hyperbolic three‐Manifolds fibering over the circle

Huang

Wang

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…To our knowledge, the only prior work on Question for non‐arithmetic hyperbolic manifolds is due to Millichap . He constructed a sequence of knots

K_{n}, K_{n}^{μ} \subset S^{3}

with incommensurable complements, such that their volumes grow linearly with

n

and such that

S^{3} ∖ K_{n}

and

S^{3} ∖ K_{n}^{μ}

share the same

n

shortest geodesics.…”

Section: Introductionmentioning

confidence: 99%

Spectrally similar incommensurable 3‐manifolds

Futer

Millichap

2017

Proc. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. 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 portion of the length spectrum but are not commensurable. More precisely, for every n 0, we construct a pair of incommensurable hyperbolic 3-manifolds Nn and N µ n whose volume is approximately n and whose length spectra agree up to length n.Both Nn and N µ n are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that these manifolds have a very thick collar about an essential surface. The two gluing maps differ by a hyper-elliptic involution along this surface. Our proof also involves a new commensurability criterion based on pairs of pants.

show abstract

“…In the non-arithmetic setting (i.e., when neither M 1 nor M 2 are arithmetic), the relationship between the geodesic length spectrum and commensurability class of the manifold is rather mysterious. To our knowledge, the only explicit work in this area is Millichap [11] and Futer-Millichap [4]. In [4], which extends work from [11], Futer and Millichap produce, for every m ≥ 1, infinitely many pairs of non-commensurable hyperbolic 3-manifolds which have the same volume and the same m shortest geodesic lengths.…”

Section: Introductionmentioning

confidence: 99%

“…To our knowledge, the only explicit work in this area is Millichap [11] and Futer-Millichap [4]. In [4], which extends work from [11], Futer and Millichap produce, for every m ≥ 1, infinitely many pairs of non-commensurable hyperbolic 3-manifolds which have the same volume and the same m shortest geodesic lengths. Additionally, they give an upper bound on the volume of their manifolds as a function of m. Inspired by [4], in this paper we consider the following question.…”

Section: Introductionmentioning

confidence: 99%

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

Linowitz

McReynolds

Pollack

et al. 2017

Comptes Rendus. Mathématique

View full text Add to dashboard Cite

ABSTRACT. In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any k ≥ 2, we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.

show abstract

Mutations and short geodesics in hyperbolic 3-manifolds

Cited by 18 publications

References 27 publications

Complex length of short curves and Minimal Fibrations of hyperbolic three‐Manifolds fibering over the circle

Complex length of short curves and Minimal Fibrations of hyperbolic three‐Manifolds fibering over the circle

Spectrally similar incommensurable 3‐manifolds

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

Contact Info

Product

Resources

About