An Upper Bound for the Volumes of Complements of Periodic Geodesics

Bergeron, Maxime; Pinsky, Tali; Silberman, Lior

doi:10.1093/imrn/rnx231

Cited by 11 publications

(25 citation statements)

References 18 publications

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“…The proof of this result for any hyperbolic metric follows from the fact that any pair of hyperbolic metrics on a hyperbolic surface are bi-Lipschitz (see, for example, [3,Lemma 4.1]).…”

Section: Coding Filling Geodesics On Surfaces By Splitting Along a Simentioning

confidence: 99%

“…In this section, we prove one of our main results, Theorem 1.1, which provides a way of constructing sequences of closed geodesics of increasing length, whose canonical lift complements are not homeomorphic and whose volumes are universally bounded by the volume of a link complement on

P T^{1} Σ

. Although this kind of examples already existed in the literature (see [3, Subsection 3.2] or [6, Example 5.2]), we point out that our method generalizes the previous examples. Theorem Given a hyperbolic surface

normalΣ

, there exist a constant

V_{0} > 0

and a sequence

{γ_{n}}

of filling closed geodesic on

normalΣ

, with

M_{{\hat{γ}}_{n}} ≇ M_{{\hat{γ}}_{k}}

for every

k \neq n

, such that

Vol false(M_{{\overset{γ}{true}}_{n}} false) < V_{0}

for every

n \in N

.…”

Section: Sequences Of Closed Geodesics With Uniformly Bounded Volume mentioning

confidence: 99%

“…Bergeron, Pinsky and Silberman gave already in [3,Example 3.1] examples satisfying Theorem 1.1 in the case of the modular surface. Their examples and the ones in the proof of Theorem 1.1 are different.…”

Section: Introductionmentioning

confidence: 99%

“…We start, however, by commenting on upper bounds for the volume of

M_{\hat{γ}}

. Bergeron, Pinsky and Silberman have already studied in [3] the problem of finding an upper bound, by giving one which is linear in the length of the geodesic. Nevertheless, it is easy to construct sequences of closed geodesics with length approaching to infinity but whose associated canonical lift complements have uniformly bounded volume.…”

Section: Introductionmentioning

confidence: 99%

“…To explain why this is the case we recall that in the modular surface [3, Section 3] they found an upper bound which is proportional to the period plus the sum of the logarithms of the coefficients corresponding to the geodesic's continued fraction expansion. The examples provided in [3, Example 3.1] have unbounded length but bounded period. On the other hand, the examples used to prove Theorem 1.1 have unbounded period.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A lower bound for the volumes of complements of periodic geodesics

Rodríguez‐Migueles

2020

J. London Math. Soc.

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“…The proof of this result for any hyperbolic metric follows from the fact that any pair of hyperbolic metrics on a hyperbolic surface are bi-Lipschitz (see, for example, [3,Lemma 4.1]).…”

Section: Coding Filling Geodesics On Surfaces By Splitting Along a Simentioning

confidence: 99%

P T^{1} Σ

normalΣ

, there exist a constant

V_{0} > 0

and a sequence

{γ_{n}}

of filling closed geodesic on

normalΣ

, with

M_{{\hat{γ}}_{n}} ≇ M_{{\hat{γ}}_{k}}

for every

k \neq n

, such that

Vol false(M_{{\overset{γ}{true}}_{n}} false) < V_{0}

for every

n \in N

.…”

Section: Sequences Of Closed Geodesics With Uniformly Bounded Volume mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We start, however, by commenting on upper bounds for the volume of

M_{\hat{γ}}

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A lower bound for the volumes of complements of periodic geodesics

Rodríguez‐Migueles

2020

J. London Math. Soc.

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Chaos and integrability in -geometry

Bolsinov

Веселов

2021

Russ. Math. Surv.

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Untitled

2023

Trans. Amer. Math. Soc. Ser. B

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Given a filling primitive geodesic curve in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the curve's canonical lift to the projective tangent bundle. In this paper we give the first known lower bound for the volume of these manifolds in terms of the length for a collection of curves with asymptotic density one. We show that estimating the volume from below can be reduced to a counting problem in the unit tangent bundle and solve it by applying an exponential multiple mixing result for the geodesic flow.

show abstract

An Upper Bound for the Volumes of Complements of Periodic Geodesics

Abstract: A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer. We give an upper bound for this volume which is linear in the geometric length of the geodesic.

Cited by 11 publications

References 18 publications

A lower bound for the volumes of complements of periodic geodesics

A lower bound for the volumes of complements of periodic geodesics

Chaos and integrability in -geometry

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