Pith. sign in

REVIEW 3 major objections 3 minor 9 references

Volumes of end-periodic mapping tori

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The volume of an end-periodic mapping torus is controlled, up to constants, by how fast the map translates on the pants graph.

desk verdict A clear, well-illustrated survey of the authors' own volume bounds for end-periodic mapping tori; no new theorems, but a useful entry point that deserves a serious referee. read the letter →

arxiv 2508.14244 v1 pith:GICDQ6IB submitted 2025-08-19 math.GT math.DS

Marissa Loving This is my paper · ORCID
classification math.GTmath.DS MSC 57K32
keywords end-periodichomeomorphisminfinite-typesurfacemappingtorushyperbolicvolumepantsgraphtranslationlengthacylindrical3-manifoldconvexmetric
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This expository paper presents an infinite-type analogue of the classical theorem that the hyperbolic volume of a pseudo-Anosov mapping torus is comparable to the monodromy's translation length on the pants graph. For a strongly irreducible end-periodic homeomorphism of an infinite-type surface, the volume of its compactified mapping torus is trapped between two positive constants times the map's asymptotic pants-graph translation length. The upper constant is universal, equal to the volume of a regular ideal octahedron, while the lower constant depends only on a quantity called the capacity of the map. The paper sketches how well-pleated surfaces supply the lower bound and how pants-block decompositions plus Dehn filling supply the upper bound. A reader should care because it gives a broad infinite-type family where dynamics on a combinatorial graph predicts the actual geometry of a hyperbolic 3-manifold.

What carries the argument

The compactified mapping torus Mbar_f is built by adding the f-invariant escaping neighborhoods U+ and U- as boundary surfaces to the infinite cyclic cover of the mapping torus and then quotienting. Its role is to turn an infinite-type surface mapping torus into a compact 3-manifold with boundary; strong irreducibility makes it acylindrical, so it carries a unique convex hyperbolic metric with totally geodesic boundary, and volume is measured in that metric. The volume-to-dynamics comparison is carried by two auxiliary devices: the capacity, defined as the pair (core characteristic, end complexity) that replaces finite-type Euler characteristic, and well-pleated surfaces, which produce bound

What would settle it

Take one of the explicit strongly irreducible end-periodic homeomorphisms built in the paper's Section 3.3, compute the exact hyperbolic volume of its compactified mapping torus by triangulating the drilled model, and compare it with Voct·τ(f) and c2·τ(f). A single such map with volume outside those bounds would falsify Theorem 4.1; matching the bounds across a family would corroborate it.

Watch

Extended reading notes

Core claim

The central object is the compactified mapping torus Mbar_f of an end-periodic homeomorphism f of an infinite-type surface. The theorem presented states that when f is strongly irreducible, the hyperbolic volume of Mbar_f is at most Voct·τ(f), where Voct is the volume of a regular ideal octahedron and τ(f) is the asymptotic translation length of f on the pants graph, and at least c2·τ(f), where c2 is a constant depending only on the capacity of f. This is put forward as the infinite-type analogue of the classical finite-type theorem for pseudo-Anosov mapping tori. The paper sketches the two mechanisms: for the lower bound, well-pleated surfaces adapted to a minimal core supply bounded-length

Load-bearing premise

The load-bearing premise is strong irreducibility: the absence of any simple closed curve or bi-infinite line whose iterates repeat in a controlled way, because this is what makes the compactified mapping torus acylindrical and gives the unique convex hyperbolic metric whose volume the theorem measures.

Editorial extensions

If this is right

  • For strongly irreducible end-periodic homeomorphisms, hyperbolic volume is coarsely determined by a purely combinatorial quantity: the pants-graph translation length.
  • The lower constant depends on capacity, so the theorem does not claim a universal lower bound across all infinite-type surfaces, unlike the universal octahedral upper bound.
  • Passing to positive powers f^N preserves the bounds because both volume and translation length scale linearly in N.
  • The upper-bound proof produces an explicit model manifold built from pants blocks, giving a concrete combinatorial handle on the geometry of Mbar_f.
  • The result resolves, in a restricted but nontrivial setting, the guiding question of how the dynamics of a surface homeomorphism predict the geometry of its mapping torus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I infer that the true asymptotic ratio of volume to pants-graph translation length, if it exists, should be expressible in terms of the end-periodic analogue of measured foliation data, and the octahedral upper constant hints at a universal maximal volume density per pants-graph step.
  • I infer that the theorem may extend to atoroidal end-periodic homeomorphisms that are not strongly irreducible, provided the cylinders coming from periodic lines are handled separately, since the proof's obstruction is specifically acylindricity.
  • I infer that the capacity-dependent lower constant is likely not sharp, and the explicit handle-shift-plus-pseudo-Anosov examples in the paper's Section 3.3 are natural test cases for computing optimal constants.
  • I infer that combining this block model with pseudo-Anosov flow representatives of end-periodic maps, as mentioned in the paper's final section, could convert these volume bounds into growth-rate bounds for the flows, though that connection is not established here.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This expository paper, based on lecture notes from a 2023 minicourse, gives an intuition- and illustration-driven overview of two recent results on volumes of end-periodic mapping tori: the upper bound of Field--Kim--Leininger--Loving [FKLL23] and the lower bound of Field--Kent--Leininger--Loving [FKLL]. The central theorem, Theorem 4.1, states that for a strongly irreducible end-periodic homeomorphism f of an infinite-type surface, the volume of the compactified mapping torus M_f satisfies c2·τ(f) ≤ Vol(M_f) ≤ c1·τ(f), where c1 is the volume of a regular ideal octahedron, c2 depends only on the capacity of f, and τ(f) is the asymptotic translation length on the pants graph. The paper develops the necessary background on infinite-type surfaces, end-periodic homeomorphisms, and depth-one foliations; explains the structure of the compactified mapping torus; sketches the lower bound via well-pleated surfaces and an interpolation; and sketches the upper bound via pants-block decompositions and Dehn filling. Full proofs are deferred to the cited papers, as the Preface explicitly states.

Significance. If the cited theorems are correct, this is a valuable didactic resource: it captures the main ideas and constructions behind a nontrivial infinite-type analogue of Brock's theorem, with helpful figures and a clear explanation of why strong irreducibility (via acylindricity) and capacity play the roles they do. The exposition is honest about its limitations: the Preface states that detailed proofs are omitted and points to the original sources. I found no hidden circularity or fitted parameters; the statement of Theorem 4.1 is consistent with the cited theorems and with the internal sketch. The main weaknesses are local mathematical slips in the proof sketches, detailed below. They do not cast doubt on Theorem 4.1 itself, because that theorem is cited from [FKLL23] and [FKLL], but they should be corrected for the paper to serve its intended expository purpose.

major comments (3)
  1. [§4.3, Lemma 4.3] The proof sketch of Lemma 4.3 is internally inconsistent with the definition of 'minimal core' given in §4.2. Removing the annular neighborhood N_ε(∂Y) from Y yields a surface homeomorphic to Y, so |χ(Y−N_ε(∂Y))| = |χ(Y)|, not < |χ(Y)|. As written, the claimed contradiction to minimality does not follow. The intended argument may use a different notion of minimal core (e.g., inclusion-minimal) or a different measure of size; please clarify, and align the definition of χ(f) and 'minimally well-pleated' with the proof in [FKLL, Lemma 4.1].
  2. [§5.2, Proposition 5.2] The block volumes in Proposition 5.2 are inconsistent with the surrounding text and with Corollary 5.3. The text states that each B_S block contributes twice as much volume as a B_T block, and Corollary 5.3 computes Vol = V_oct(n_T + 2n_S). Proposition 5.2 as printed gives B_S volume V_oct and B_T volume 2V_oct, and the metric subscripts σ_BT/σ_BS are interchanged. Please correct the statement so the numerical assignments match the path-metric edge lengths.
  3. [§5.5, Step 2] The displayed inequality τ(f,Ω) ≤ τ(f) − ε is false under the hypothesis |τ(f)−τ(f,Ω)| < ε; the correct bound is τ(f,Ω) < τ(f)+ε. The epsilon argument for the upper bound works with the corrected sign. Please fix this before publication.
minor comments (3)
  1. [§4.4, Step 4] The symbol K is used both for the uniform power K ≥ 5|χ(f)| and for the Brock-type constant appearing in the final inequality of the lower-bound sketch. Rename one of these (e.g., use C for the constant) so that the statement 'where K relies only on the capacity of f' is unambiguous.
  2. [§5.5, Steps 1–2] The quantity τ(f,Ω) is used but never defined. Please add the definition: the asymptotic translation length of f restricted to the f-invariant component Ω of the pants graph P(S).
  3. [Various] Minor typos: 'psuedo-Anosov' in Theorem 2.5; 'aclyindricity' in the paragraph after Proposition 3.8; 'Handell-Miller' in Section 6. Also, the Preface's explicit statement that full proofs are omitted is appropriate and should remain.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is an expository survey whose central theorem is imported from independent prior work, with no fitted parameters or definitional reductions.

full rationale

The manuscript explicitly identifies itself as an expository overview (Preface, §1) of the main theorems of [FKLL23] and [FKLL]. Theorem 4.1 is presented as the combination of those two papers' results, not derived from data or from a parameter fitted in this text. The load-bearing notions—strong irreducibility (Definition 3.4), capacity (χ(f), ξ(f)), and τ(f)—are defined independently of the volume bounds, and no equation of the paper defines one side of Theorem 4.1 in terms of the other. The sketched proofs rely on standard external theorems (Richards classification, Thurston hyperbolization, Mostow rigidity, Roussarie–Thurston normal form, Agol’s block-volume computation, Thurston’s Dehn surgery theorem) and on the cited papers for the substantive estimates (Propositions 3.7–3.8, Lemma 4.3, Claim 5.7). Self-citation is central to the survey format, but the cited works are independent mathematical results with explicit hypotheses; Brock’s theorem is invoked as an external finite-type analogue, not as an input to the bound. The Preface’s caveat that proofs are not fully reproduced here is a limitation of the survey genre, not evidence of circularity. No prediction reduces by construction to its input, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. The constants c1, c2, K, beta are defined from established theory and topology; they are not fitted to data. The axioms listed are the standard background results invoked in the proof sketches.

assumptions (6)
  • standard math Roussarie-Thurston Normal Form (Theorem 2.9)
    Used in the proof sketches of Propositions 3.7 and 3.8 to put tori and annuli into normal form relative to the foliation.
  • standard math Thurston Hyperbolization Theorem (Theorem 2.4) and fibered case (Theorem 2.5)
    Used to establish existence of hyperbolic metrics on M_f and its double, and to characterize when mapping tori are hyperbolic.
  • standard math Basmajian's tubular neighborhood theorem [Bas94]
    Used in Lemma 4.3 to guarantee a product neighborhood of length beta for the boundary of M_f, giving the bound on the length of the core boundary.
  • standard math Agol's volume computation for pants blocks [Ago03] (Proposition 5.2)
    Used to compute the volume of the model manifold in the upper bound proof, yielding the constants in terms of the ideal octahedron volume.
  • domain assumption Strong irreducibility of end-periodic homeomorphisms (Definition 3.4)
    The main theorem is only stated for such homeomorphisms; strong irreducibility (no reducing curves, no periodic lines, no periodic curves) is assumed throughout and used to prove acylindricity.
  • domain assumption Capacity (chi(f), xi(f)) is finite and well-defined
    The lower bound constant c2 depends on the capacity; the definition requires f to have finitely many ends all accumulated by genus, which is part of the standing hypothesis.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Volumes of end-periodic mapping tori." pith.science (2026). https://pith.science/paper/GICDQ6IB

@misc{pith2026250814244,
  author       = {Pith},
  title        = {Pith review of: Volumes of end-periodic mapping tori},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GICDQ6IB}},
  note         = {Machine review of arXiv:2508.14244}
}
read the original abstract

In this expository paper, we provide an intuition and illustration-driven overview of two recent results that tie the dynamics of certain homeomorphisms of infinite-type surfaces, called end-periodic homeomorphisms, to the geometry of their associated (compactified) mapping tori. These results are analogues of a theorem of Brock in the finite-type setting for mapping tori of pseudo-Anosov homeomorphisms.

Figures

Figures reproduced from arXiv: 2508.14244 by the authors.

Figure 1
Figure 1. Two infinite-type surfaces with end spaces homeomorphic to a Cantor set [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. An exiting sequence on the ladder surface. The interested reader can refer to a survey of big mapping class groups by Aramayona and Vlamis for a more in-depth discussion of infinite-type surfaces [AV20]. 2.2. Homeomorphisms of infinite-type surfaces. The most straightforward examples of homeomorphisms of infinite-type surfaces are those that directly correspond to homeomorph￾isms of finite-type surfaces such as rota… view at source ↗
Figure 3
Figure 3. It is a consequence of the classification theorem that these three surfaces are homeomorphic [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (31 more)
Figure 4
Figure 4. Figure 4: A handle strip and its image after applying the associated shift map. call h, is a handle shift. Note that the handle shift h : S → S is only of intrinsically infinite type when T is embedded between two distinct ends of S. Just as in the finite-type setting, we can st…
Figure 5
Figure 5. Figure 5: A foliation of a torus by simple closed curves, a foliation of the torus coming from a line of irrational slope (note that it has no closed leaf), and a pair of singular, transverse foliations on a genus 2 surface. 2.3. Foliations of 3-manifolds. A foliation of an n-di…
Figure 6
Figure 6. Figure 6: The mapping torus of a homeomorphism f : Σ → Σ. Note that Mf only depends on the mapping class of f [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: An infinite geodesic accumulating onto a simple closed geodesic on a surface. However, an even more pressing question in the reader’s mind might be, how do these foliations relate to the infinite-type surfaces discussed at the beginning of this section? The following r…
Figure 8
Figure 8. Figure 8: A depth one foliation of a closed 3-manifold with a single, compact genus 2 leaf. Note that for a closed 3-manifold, a compact leaf has finite genus without boundary components and a non-compact leaf has infinite genus without planar ends or boundary components. Why ar…
Figure 9
Figure 9. Figure 9: A cross section of the Reeb foliation of the solid torus. Corollary 2.11 follows quite directly from the following characterization of Reebless folia￾tions. Theorem 2.12 (Novikov, [Nov65]). Let M be a 3-manifold and let F be a Reebless foliation. Then (1) every leaf λ …
Figure 10
Figure 10. Figure 10: Three surfaces with finitely many ends all accumulated by genus [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: The name end-periodic describes the fact that outside of some compact subsurface the action of f : S → S is periodic in certain neighbor￾hoods of the ends of S. 3.2. Geometry of end-periodic mapping tori. In this section we pose and then address the following question…
Figure 12
Figure 12. Figure 12: The end-periodic mapping tori Mf and its double DMf , The first part of Theorem 3.2, establishing the existence of Mf , is well-known (see [Fen97]), while Theorem 3.2 (i) follows from Theorem 2.4 and Theorem 3.2 (ii) follows from the following straightforward argument…
Figure 13
Figure 13. Figure 13: A reducing curve which is sent from a neighborhood of a repelling end into a neighborhood of an attracting end. (2) AR-periodic lines, i.e. a line ℓ with one end in a nesting neighborhood of an attracting end and the other end in a nesting neighborhood of a repelling …
Figure 14
Figure 14. Figure 14: An AR-periodic line is shown in purple and a periodic line is shown in pink. The green and blue arrows indicate the direction the end￾periodic homeomorphism is shifting in the ends. (3) periodic curves, i.e. a simple closed curve γ such that f n (γ) = γ for some n ̸= …
Figure 15
Figure 15. Figure 15: ) [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]
Figure 16
Figure 16. Figure 16: The subsurface C (shown in blue) is separating and the subsur￾face C ′ (shown in brown) is fully separating with respect to the handle shifts shown in green. In [FKLL23], we introduced the notion of strong irreducibility, which disallows any periodic lines including t…
Figure 17
Figure 17. Figure 17: Mf shown as a compactification of Mf with boundary surfaces S+ and S−. Let U− be the union of nesting neighborhoods of the repelling ends and let U− = S n≥0 f n (U−). We call U− the negative escaping set for f. In the same way define the positive escaping set for f to…
Figure 18
Figure 18. Figure 18: An incompressible torus giving rise to a periodic curve. Proposition 3.8 ([FKLL23]). If f : S → S is a strongly irreducible end-periodic homeo￾morphism, then Mf is acylindrical. The argument for aclyindricity is more subtle than the one we used to show that Mf is ator…
Figure 19
Figure 19. Figure 19: The cylinder on the left is labeled “AR-periodic lines and reduc￾ing curves” because it could arise from either phenomenon since it runs from a component of ∂Mf corresponding to a repelling end of f : S → S to a com￾ponent of ∂Mf corresponding to an attracting end of …
Figure 20
Figure 20. Figure 20: Two vertices in the pants graph are connected by an edge if their associated pants decompositions are related by one of the two elementary moves shown here. 4. A lower bound on volume Given a taut depth-one foliation we are now prepared to address Question 2.8! In par…
Figure 21
Figure 21. Figure 21: Let S be the ladder surface. Then P and P ′ are in different connected components of P(S). pants decompositions which are in different connected components of P(S). Thus, we really are taking this infimum over the union of connected components for which d(P, f n (P)) …
Figure 23
Figure 23. Figure 23: Why do we pass to Σ × R? Brock understands this picture very well since it puts us in the setting of convex cores of quasi-Fuchsian hyperbolic 3-manifolds [Bro03a]. If we start with a bounded length pants decomposition on the initial surface, then this interpolation p…
Figure 22
Figure 22. Figure 22: Recall that in a surface the collar lemma tells us that a short simple closed curve lives in a collar with definite area. The analogue of this in 3-dimensions is the Margulis lemma which tells us that a short curve in a 3-manifold is contained in a solid torus, called…
Figure 23
Figure 23. Figure 23: The interpolation through simplicial hyperbolic surfaces is shown by the crumpled blue surfaces in Σ × R where the covering transformation is given by F(x, t) = (f(x), t + 1) [PITH_FULL_IMAGE:figures/full_fig_p018_23.png]
Figure 24
Figure 24. Figure 24: The homeomorphism f = hρ is strongly irreducible so long as ρ acts with sufficiently large translation length on the curve complex of the subsurface C. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_24.png]
Figure 25
Figure 25. Figure 25: An illustration (drawn down a dimension) of the parts of a fiber S in Mf that can’t be pushed into ∂Mf . On the other hand, “how big is C?” is not quite the right thing to measure! Instead we need to ask the following, 2’) what is the size of the smallest subsurface o…
Figure 26
Figure 26. Figure 26: Note that you can go from a triangulation or arc system to a lamination by “spinning”. If λ is the smallest geodesic lamination satisfying (2) and (3), we call it the pleating locus for φ. In [FKLL], we introduced the notion of a well-pleated surface as a natural adap…
Figure 27
Figure 27. Figure 27: The covering transformation in both the finite-type and infinite￾type setting is given by F(x, t) = (f(x), t − 1). εℓσ(∂Y ), by some basic hyperbolic trigonometry [Bus10]. Since Area(Nε(∂Y )) ≤ Area(Y, σ), then εℓσ(∂Y ) ≤ 2π|χ(Y )|. Thus, if ℓσ(∂Y ) > 2π|χ(Y )| β , th…
Figure 28
Figure 28. Figure 28: On the left is an illustration of the interpolation through pleated surfaces in Mf∞. Note that as we move through this interpolation we are “unzipping” along the top boundary of Mf∞ and “zipping up” along the bottom boundary of Mf∞. Hence, the subsurface of S where we…
Figure 29
Figure 29. Figure 29: On the top is the pants block B T obtained as a quotient of Σ1,1×I and on the the bottom is the pants block B S obtained as a quotient of Σ0,4×I. 5. An upper bound on volume Our approach to proving the upper bound in Theorem 4.1 will follow a proof of Brock’s upper bo…
Figure 30
Figure 30. Figure 30: The boundaries of B˚S and B˚T both consist of a disjoint union of three-holed spheres (i.e. pairs of pants) as shown. collection of boundary components of three-holed spheres shown in [PITH_FULL_IMAGE:figures/full_fig_p024_30.png]
Figure 31
Figure 31. Figure 31: Realizing the path P = P0, P1, . . . , Pn = f −1 (P) in MF . Recall that S-moves and T-moves are the elementary moves associated to edges in the pants graph. 5.3. Building our model manifold. Our main strategy for building the model manifold Mcf is to first attempt to…
Figure 32
Figure 32. Figure 32: A cartoon illustrating the collapsing of flow lines of Ψs under h. Since Mf−PΩ is acylindrical, it admits a convex hyperbolic structure with totally geodesic, thrice punctured sphere boundary components. Idea of proof: Time to flow! There is a suspension flow (ψs) on …
Figure 33
Figure 33. Figure 33: An illustration of gluing a solid torus D × S 1 into the torus boundary component T of N − γ with a specified choice of Dehn filling coeffi￾cient. This process of drilling and filling can be extended to a link L ⊂ N, which is the image of an embedding of a disjoint un…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

  1. [3]

    Examples of endperiodic automorphisms

    29 [CC16] John Cantwell and Larwence Conlon. Examples of endperiodic automorphisms. Preprint, arXiv:1008.2549,

    work page Pith review arXiv

  2. [1923]

    [LMT] Michael Landry, Yair Minsky, and Samuel J. Taylor. Endperiodic maps via pseudo-Anosov flows. arXiv:2304.10620. [LMT23] Michael Landry, Yair Minsky, and Samuel J. Taylor. Flows, growth rates, and the veering poly- nomial. Ergodic Theory and Dynamical Systems , 43(9):3026–3107,

    work page Pith review arXiv

  3. [1974]

    Stallings

    [Sta78] John R. Stallings. Constructions of fibred knots and links. In Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2 , Proc. Sympos. Pure Math., XXXII, pages 55–60. Amer. Math. Soc., Providence, R.I.,

    work page 1976

  4. [1986]

    Short curves of end-periodic mapping tori

    [Whi] Brandis Whitfield. Short curves of end-periodic mapping tori. arXiv:2408.07044. 31

    work page Pith review arXiv

  5. [1997]

    A lower bound on volumes of end-periodic mapping tori

    [FKLL] Elizabeth Field, Autumn Kent, Christopher Leininger, and Marissa Loving. A lower bound on volumes of end-periodic mapping tori. arXiv:math.GT/2306.03279. [FKLL23] Elizabeth Field, Heejoung Kim, Christopher Leininger, and Marissa Loving. End-periodic home- omorphisms and volumes of mapping tori. J. Topol., 16(1):57–105,

    work page Pith review arXiv

  6. [2001]

    Translated from the 1996 French original by Leslie D. Kay. [Par] Hugo Parlier. A shorter note on shorter pants. arXiv:2304.06973. [Ric63] Ian Richards. On the classification of noncompact surfaces. Trans. Amer. Math. Soc., 106:259–269,

    work page Pith review arXiv 1996

  7. [2003]

    Infinite-type loxodromic isometries of the relative arc graph

    [AMP21] Carolyn R. Abbott, Nicholas Miller, and Priyam Patel. Infinite-type loxodromic isometries of the relative arc graph. arXiv:math.GT/2109.06106,

    work page Pith review arXiv

  8. [2007]

    Thurston

    [Thu] William P. Thurston. Hyperbolic structures on 3-manifolds, ii: Surface groups and 3-manifolds which fiber over the circle. arXiv:1610.02592. [Thu72] William P. Thurston. Foliations of 3-manifolds that are circle bundles . PhD thesis, UC Berkeley,

    arXiv

Show all 9 references
  1. [2010]

    [Can96] Richard D

    Reprint of the 1992 edition. [Can96] Richard D. Canary. A covering theorem for hyperbolic 3–manifolds and its applications. Topology, 35(3):751–778,

    1992

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.