Quasi-geodesics in the Cannon-Thurston metric

Caglar Uyanik; Catherine Pfaff; Joseph Maher; Vaibhav Gadre

REVIEW 3 major objections 4 minor 2 cited by

Quasi-geodesics in the Cannon-Thurston metric

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The Cannon–Thurston pushforwards of surface measures — Lebesgue and random-walk hitting measures on the circle — are mutually singular with every natural 3-manifold measure, because typical geodesics spend different amounts of time near a f

desk verdict A real advance on Cannon–Thurston measure singularity, with a load-bearing gap in the 'full surface measure' statement and a small rescaling error in Prop 52. read the letter →

arxiv 2510.04350 v3 pith:3AB2PQJX submitted 2025-10-05 math.GT math.DSmath.GR

Vaibhav Gadre , Joseph Maher , Catherine Pfaff , Caglar Uyanik This is my paper · ORCID

classification math.GTmath.DSmath.GR MSC 57K3237D4060B15

keywords Cannon–Thurstonmapfiberedhyperbolic3-manifoldsmutualsingularityofmeasureshittingrandomwalksongroupsquasi-geodesicsmeasuredlaminationsgeodesicflow

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 paper proves that, for a closed hyperbolic 3-manifold fibering over the circle, the Cannon–Thurston map pushes forward the natural measures on the fiber's circle at infinity (Lebesgue measure, and hitting measures of full random walks on the surface group) to measures on the sphere at infinity that are mutually singular with the natural 3-manifold measures there (Lebesgue measure, and hitting measures of geometric random walks on the 3-manifold group). The mechanism is a geodesic-statistics dichotomy: almost every geodesic sampled by a pushed-forward surface measure spends a definite positive fraction of its length near the lifted base fiber, while almost every geodesic sampled by a 3-manifold measure spends a fraction that tends to zero. For geometric surface measures the positive fraction is upgraded to an explicit exponential rate 1 − K e^{−αkR}. The proof builds explicit quasi-geodesics in the Cannon–Thurston metric, controlled by a height function measuring distance to the invariant laminations, which may be useful beyond the measure-singularity question.

What carries the argument

The central object is the height function hθ(v) = log_k⌊log(1/d(v, Λ1+)) − log(1/θ)⌋_1 − log_k⌊log(1/d(v, Λ1−)) − log(1/θ)⌋_1 on the unit tangent bundle of the surface, where Λ1± are the extended invariant laminations. For a non-exceptional geodesic γ the test path τγ(t) = (γ(t), hθ(γ1(t))) is shown to be a uniform unparametrized quasigeodesic in the Cannon–Thurston metric with the same endpoints as ι(γ). Combined with rectangle/bottleneck estimates (optimal-height rectangles are bottlenecks for opposite quadrants), this yields the effective control of time spent near the base fiber. The non-effective half uses ergodicity of the geodesic flow and, for the 3-manifold side, a mixing estimate f

What would settle it

Compute the Cannon–Thurston distance between adjacent fibers S×{n} and S×{n+1} in the universal cover of the mapping torus. If it equals log k rather than 1, the estimate d(w_n x0, S0) ≥ 2A log n in Proposition 52 must be rescaled by log k, and for k < e the claimed 1/√n decay (and hence zero limiting fiber-time) may fail for the hitting-measure half of Theorem 8.

Watch

Extended reading notes

Core claim

Theorem 4: pushforwards of full surface measures by the Cannon–Thurston map are mutually singular with every 3-manifold measure. The proof rests on two statistical statements: Theorem 6 (a ι∗ν-typical geodesic has liminf T→∞ (1/T)|γ([0,T]) ∩ N_R(S0)| ≥ ε > 0) and Theorem 8 (a ν-typical geodesic for any 3-manifold measure has the same proportion tend to 0). The dichotomy gives an explicit description of sets witnessing singularity. Theorem 7 strengthens the surface half, for geometric surface measures, to a lower bound 1 − K e^{−αkR}.

Load-bearing premise

The random-walk half of Theorem 8 assumes adjacent fibers are exactly one unit apart in the Cannon–Thurston metric, but the metric's definition gives separation log k (k = stretch factor); the displayed estimates need rescaling that is not provided.

Editorial extensions

If this is right

Full surface measure pushforwards are mutually singular with all 3-manifold measures (Theorem 4).
Hitting measures from incompressible surfaces in closed hyperbolic 3-manifolds are singular with Lebesgue measure and with π1(M)-random-walk hitting measures (Corollary 5).
For geometric surface measures the proportion of time near the fiber is exponentially close to 1, at rate e^{−αkR} (Theorem 7).
The explicit quasi-geodesic test paths in the Cannon–Thurston metric give a uniform description of geodesic behavior that can be used for other averaging problems.
Sets witnessing singularity are described concretely: geodesics with positive limiting fiber-time versus those with zero limiting fiber-time.

Reading between the lines

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

The dichotomy likely extends to broader classes of stationary measures: if the paper's mechanism transfers, any stationary measure on the sphere whose geodesics avoid fibers would be singular to any surface-pushforward with positive fiber time.
The log k vs 1 fiber-separation issue in Proposition 52 suggests the random-walk half of Theorem 8 needs a rescaling by log k; if k<e the stated 2A log n estimate may fail, though singularity may survive with adjusted constants.
The test-path construction gives a candidate route to effective statistics for saddle connections in the singular solv metric, which the paper explicitly does not pursue.
A testable extension: for random walks on π1(M) with finite first moment but not finite exponential moment, the fiber-avoidance dichotomy should still hold if the local limit theorem is replaced by a weaker recurrence estimate.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read it closely. The main result is genuinely new: it upgrades Tukia's Lebesgue-vs-Lebesgue singularity, and the recent conformal variants, into a geometric criterion that covers random-walk hitting measures on both sides and gives explicit exponential rates (Theorem 7). The architecture is clean: Theorem 4 falls out of Theorems 6 and 8 by a full-measure-set singularity argument, and the surface-side proofs rest on standard geodesic-flow ergodicity. The height-function/quasi-geodesic machinery in Part II is substantial and appears to be doing real work, not just decoration. The paper also cites Tukia, Kim–Oh, and Kim–Zimmer properly; the novelty claim does not depend on ignoring them.

The soft spot is the definition of 'full surface measures' (Definition 2), where the random-walk estimates in §2.8 are stated under finite exponential moment. Propositions 32, 35, and Corollary 33 need it; Proposition 37 is stated for nonelementary μ, but its proof cites Lemma 36, which is only proved for geometric μ. Lemma 51 and Proposition 41 then invoke ergodicity and double ergodicity for the full class without reconciling this. That is load-bearing for Theorem 6, and it is not a cosmetic gap: for heavy-tailed μ with no finite exponential moment the tracked-geodesic times t_n can have uncontrolled gaps, so the 'linearly often' step can fail. The fix is either to restrict Theorem 6 to geometric surface measures or to prove the needed estimates for the full class. My guess is the qualitative result is true, and the argument can be repaired, but the statement as written overreaches.

There is a smaller, easily fixed error in Proposition 52: adjacent fibers in the Cannon–Thurston metric are log k apart by Eq. (1), not distance one. That only rescales the constant A, but the text should say so.

Who is this for: people working on Cannon–Thurston maps, stationary measures on boundaries, and random walks on surface/manifold groups. They will want the effective theorem and the quasi-geodesic construction regardless of the measure-class issue. I would cite it, with a caveat until the full-measure case is sorted. It deserves a serious referee: the central idea is sound, the work is honest, and the problems are repairable. Recommend conditional acceptance subject to those fixes.

Referee Report

3 major / 4 minor

Summary. The paper proves a measure-singularity theorem for Cannon–Thurston maps of closed hyperbolic fibered 3-manifolds. The main result, Theorem 4, states that the pushforward to S²∞ of any "full surface measure" on S¹∞ (including Lebesgue measure and hitting measures of nonelementary, full random walks on π₁(S)) is mutually singular with any "3-manifold measure" on S²∞ (Lebesgue measure or hitting measures of geometric random walks on π₁(M)). The proof is organized around two geodesic-statistics assertions: Theorem 6, that geodesics sampled by pushforwards of surface measures spend a definite positive proportion of time near the base fiber S₀, and Theorem 8, that geodesics sampled by 3-manifold measures spend asymptotically negligible time near S₀. The effective Theorem 7 gives exponential rates for geometric surface measures, using an explicit construction of quasigeodesic test paths in Section 6. The paper is well structured and the overall architecture—deducing singularity from incompatible typical behavior of geodesics—is sound.

Significance. If the theorem holds in the stated generality, it provides a unified geometric explanation for Tukia's Lebesgue singularity theorem and extends singularity statements to random-walk hitting measures. The construction of uniform quasigeodesic test paths with a height function and tame bottlenecks is substantial and of independent interest, and the effective exponential bounds in Theorem 7 are a genuine strengthening. The paper also correctly identifies the key geometric inputs: ladders over lamination leaves, separation properties, and quasigeodesic stability. The manuscript is careful in many places, especially in isolating the suited-lamination hypothesis and in Remark 78 concerning bounded geometry. However, the full generality of Theorem 4 currently rests on moment-free random-walk estimates that are neither stated nor proved, and the random-walk half of Theorem 8 contains a metric-scaling error; these issues need to be addressed before the central claim is fully supported.

major comments (3)

[Definition 2 and §3.3 (Lemma 51); §2.8] Theorem 4 and Theorem 6 quantify over all full surface measures, which by Definition 2 include nonelementary full probability measures with no finite exponential moment. However, the random-walk estimates used in the proof assume geometric or finite-exponential-moment measures: Proposition 32, Corollary 33, Proposition 35, and Proposition 37 are stated with finite exponential moment; Lemma 36 is stated for geometric μ; and Corollary 34's upper linear-rate conclusion also uses finite exponential moment. Lemma 51's argument that good events occur "linearly often" needs not only ergodicity of the shift map but also control of the projected times t_n in order to convert event frequencies into a positive proportion of geodesic time. For a heavy-tailed μ, t_n/n may diverge, so the conversion can fail. No moment-free version of these estimates is supplied. This is load-bearing because Theorem 4
[§3.3, proof of Proposition 52] The proof states that the distance between adjacent fibers S×{n} and S×{n+1} is equal to one in the Cannon–Thurston metric. This contradicts Eq. (1): the vertical term is (log k)² dz², so the metric distance between fibers with z-coordinates n and n+1 is log k, not 1 (unless k=e). Consequently, the implication |φ(w_n)| ≥ 2A log n ⇒ d(w_n x₀, S₀) ≥ 2A log n needs a factor 1/log k in the choice of A. The error appears repairable by rescaling A, but as written the proof contains a false metric statement in a central step of Theorem 8.
[§3, Proposition 41 and §3.3, Lemma 51] The proof that ν-almost all geodesics are non-exceptional for full surface measures invokes double ergodicity of the boundary action, citing Kaimanovich [Kai03, Theorem 17]. Double ergodicity is typically stated under finite-first-moment or finite-exponential-moment hypotheses, and the manuscript does not verify that Definition 2's full measures satisfy them. Similarly, Lemma 51's use of "ergodicity" to get linearly many translates of α along a random-walk-sampled geodesic is not tied to a specific ergodic theorem; for finite-moment measures one could use Corollaries 33–35, but those are exactly the inputs missing in the no-moment case. This is part of the same gap as the first major comment, but it is important enough to flag separately because Proposition 41 is needed already to define the test paths and to rule out exceptional geodesics.

minor comments (4)

[Title/abstract] The arXiv title is "Quasi-geodesics in the Cannon-Thurston metric," while the manuscript itself is titled "Singularity of Cannon–Thurston maps" and the abstract is correspondingly different. Please align the title and abstract for the intended submission.
[Definitions 54 and 94] The height function is defined twice, in Definition 54 and again in Definition 94, with slightly different wording (one says "regular pair," the other "suited pair"). Merge these into a single definition to avoid confusion.
[§4, Theorem 61] The effective bound Theorem 7 relies on Theorem 61, imported from [GH24], which states that d(ι(γ(0)), ι(γ(T))) grows linearly for Lebesgue-almost all γ. If [GH24] is a companion or unpublished manuscript, please state the exact theorem and, ideally, sketch the proof; as written, this is an external dependency for the effective half.
[§3.2, proof of Lemma 49] The phrase "the forward and backward projections of φ'_t to eS_h × R converge to Lebesgue measure" is vague; these are pushforwards, not projections. Please clarify the wording.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the singularity theorem is derived from independent geometric and ergodic estimates, not from its own conclusion.

full rationale

The paper's central claim (Theorem 4) is derived from two independent dynamical statements: surface-measure typical geodesics spend a definite proportion of time near the base fiber (Theorem 6) and 3-manifold-measure typical geodesics spend asymptotically negligible time near it (Theorem 8). Neither of these statements is assumed or fitted; Theorem 8 uses mixing estimates of Oh–Pan (Theorem 50) and the local central limit theorem (Proposition 53), while Theorem 6 uses random-walk hitting measure properties, Birman–Series estimates, and the explicit geometric construction of test paths. The effective Theorem 7 uses the paper's own quasigeodesic construction (Theorem 55), verified through Farb's criteria, together with Birman–Series volume bounds and a cited linear-growth result of Gadre–Hensel. That cited theorem is a self-citation only in the weak sense that one of the present authors is a coauthor of the cited work, but it is a parameter-free, externally stated prior theorem about linear growth of Cannon–Thurston distance along typical geodesics; it is not the target singularity claim and does not by construction force the conclusion. No constants are fitted to the predicted singularity, and no uniqueness theorem from the authors' prior work is invoked to exclude alternatives. The manuscript's own Remark 78 explicitly notes a limitation of the effective discussion in the bounded-geometry setting, which is a scope restriction, not a circular step. The remaining concerns about finite-exponential-moment hypotheses versus 'full surface measures' are potential correctness or gap issues, not instances where a prediction reduces to its input by definition.

Assumptions & free parameters 1 free parameters · 10 assumptions · 3 invented entities

This is a pure-math proof: there is no data and no constant fitted to a target. The single hand-chosen constant (θ_Λ) is a smallness cutoff, and the quasigeodesicity theorem is proved rather than engineered by the choice. All other inputs are named theorems from the literature (CT07, BF92, Mitra, Hopf, Oh–Pan, BMSS23/Gou22, LL10, BS85, CG06/Can96). The genuinely new objects — height function, test paths, extended laminations, segment decomposition — are internally validated within the paper. The ledger shows the result rests on a large stack of deep cited theorems, which is normal for this field but places the burden of verification on the citations.

free parameters (1)

θ_Λ (height-function cutoff) = θ_Λ = θ_min^6 · e^{−6(T0 + LΛ + 3ρΛ + DΛ + QΛ cΛ)} (Definition 109)
A hand-chosen smallness constant in the height function (Defs 54/94/109). It is a technical cutoff controlling the distance to the extended laminations; it is not fitted to the singularity conclusion and affects only the effective rates, not the qualitative theorems.

assumptions (10)

domain assumption Cannon–Thurston metric is quasi-isometric to the hyperbolic metric on H³ [CT07, Thm 5.1] (Theorem 12).
Invoked throughout: boundary measures, geodesics and quasigeodesics in eS_h×R are transferred to H³. Load-bearing for comparing the two measure families.
domain assumption The Cannon–Thurston metric is Gromov hyperbolic via the Bestvina–Feighn combination theorem [BF92, p.88] (Theorem 13).
Enables δ-hyperbolic tools: quasiconvexity, Morse lemma, fellow-traveling, Farb's quasigeodesic criterion.
domain assumption Ladders over geodesics in eS_h are K-quasiconvex in eS_h×R [Mit98, Lemma 4.1] (Theorem 14).
Keeps target geodesics of surface geodesics inside a bounded neighborhood of the ladder F(γ); foundation of the Section 6 arguments.
standard math Ergodic uniform distribution of Lebesgue geodesics in T¹(M) (Prop 28, Hopf).
Source of the positive proportion of time near a fixed closed geodesic used in Lemma 48 for Lebesgue-typical surface geodesics.
domain assumption Oh–Pan mixing theorem for Z-covers of a hyperbolic 3-manifold [OP19, Thm 1.7] (Theorem 50).
Gives the t^{-1/2} decay of correlation used to show Lebesgue 3-manifold geodesics spend negligible time near the base fiber (Lemma 49).
domain assumption Random-walk estimates: linear progress, exponential shadow decay, gap bounds [BMSS23], [Gou22], [Kai94] (Props 32–37).
Backbone of the hitting-measure cases of Theorems 6 and 8. Stated for geometric/finite-exponential-moment measures, while Definition 2 also admits 'full' non-geometric measures; the paper does not reconcile this.
standard math Local Central Limit Theorem for aperiodic random walks on Z [LL10, Prop 2.4.4] (Proposition 53).
Supplies the 1/√n decay of the probability that the projected walk φ(w_n) lies in an O(log n) window around zero, used in Proposition 52.
domain assumption Birman–Series bounds on neighborhoods of geodesic laminations [BS85] (Theorems 58 and 64).
Produces the r²(log 1/r)^{6g−6} unit-tangent volume bound and the square-covering bound for the limit set; these yield the exponential rate in Theorem 7.
domain assumption Double ergodicity of the boundary action for hitting measures of random walks on surface groups [Kai03, Thm 17].
Used in Proposition 41 to show almost every geodesic is non-exceptional. Cited, not proved, and its usual hypotheses are stronger than Definition 2's bare 'full' condition.
domain assumption Tameness and Covering theorems: incompressible surfaces in hyperbolic 3-manifolds are quasi-Fuchsian or virtually fibered [CG06], [Can96].
Powers Corollary 5, extending the singularity statement to all incompressible surfaces.

invented entities (3)

Height function h_θ and test path τ_γ
purpose: Straighten exponentially distorted images ι(γ) of surface geodesics into uniform (Q,c)-quasigeodesics in the Cannon–Thurston metric; the technical engine for the effective bounds (Theorem 7).
Introduced in Definitions 54/94; validated only by the paper's own Theorem 55 proof via Farb's criteria. No external falsifiable handle; a mathematical construction whose correctness carries the main residual risk of Part II.
Extended laminations Λ̄⁺, Λ̄⁻ (extended leaves added in complementary regions)
purpose: Provide a well-behaved distance-to-lamination function on T¹(S_h) so the height function is uniform inside non-triangular ideal polygons.
Standard 'add leaves inside ideal polygons' device (Definition 71); properties proved in Propositions 72–75. Internally validated; not an independently evidenced entity.
Corner/straight segment decomposition and tame bottlenecks
purpose: Split a geodesic into pieces that are bottlenecks (corner segments) or locally quasigeodesic (straight segments), enabling Farb's uniform-progress argument.
Technical decomposition (Definitions 83–84; Lemmas 100–101) used to prove uniform quasigeodesicity; internal to the paper only.

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Pith. "Pith review of Quasi-geodesics in the Cannon-Thurston metric." pith.science (2026). https://pith.science/paper/3AB2PQJX

@misc{pith2026251004350,
  author       = {Pith},
  title        = {Pith review of: Quasi-geodesics in the Cannon-Thurston metric},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AB2PQJX}},
  note         = {Machine review of arXiv:2510.04350}
}

read the original abstract

A closed fibered 3-manifold admits a complete hyperbolic metric if and only if it has a fibration with a pseudo-Anosov monodromy. The stable and the unstable laminations associated to the pseudo-Anosov homeomorphism on the fiber surface give rise to a natural metric on the 3-manifold, the Cannon-Thurston metric, which is quasi-isometric to the hyperbolic metric. In this paper, we describe a specific family of quasi-geodesics in the Cannon-Thurston metric. We use the main results of this article in a companion paper to obtain statistics for typical geodesics with respect to various natural measures on the 2-sphere, thus giving a geometric criterion for singularity between some of these measure classes.

Figures

Figures reproduced from arXiv: 2510.04350 by the authors.

**Figure 2.** Ladders over leaves are quasi-isometric to [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Notation for the complements of the leaves [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

Figures from the paper (24 more)

**Figure 4.** Figure 4: Extended leaves in a complementary region with four sides. [PITH_FULL_IMAGE:figures/full_fig_p042_4.png]

**Figure 5.** Figure 5: Two ideal polygons intersecting in a non-rectangular polygon. [PITH_FULL_IMAGE:figures/full_fig_p044_5.png]

**Figure 6.** Figure 6: Opposite quadrants. Since all sides of R have positive measure, the four leaves containing the sides of R have distinct endpoints at infinity (no two are asymptotic). They divide Seh into nine complementary regions, of which only R is compact. We call a (non-compact) c…

**Figure 7.** Figure 7: Leaves of Λ− intersecting a common leaf of Λ+. We now show that there is a lower bound on the measure of an (ℓ1, ℓ2) maximal rectangle. Proposition 81. Suppose that (Sh,Λ) is a hyperbolic metric on S together with a suited pair of measured laminations. Then there is a …

**Figure 8.** Figure 8: Intersecting geodesics. In particular, min{θ1, θ2} ⩾ 1 2 θ, and up to relabeling, we may assume that θ1 ⩾ 1 2 θ. Let d = d(x, y). Using the sin rule for right angled triangles in H2 , sin θ1 = sinh r1 sinh d . which we may rewrite as sinh d = sinh r1 sin θ1 . We will u…

**Figure 9.** Figure 9: An absolute value function. With this notation, we may rewrite (14) as |t|Eℓ − K ⩽ ργ,ℓ(t) ⩽ |t|Eℓ + K, (15) where K = log L0. We now use the above observations to show that if the value of the radius function ργ,ℓ(t) is sufficiently large, then there is an interval ce…

**Figure 10.** Figure 10: A transverse rectangle. A choice of unit speed parametrization for a geodesic γ orders the leaves of the laminations intersecting γ. Using the ordering and our conventions for rectangles illustrated in [PITH_FULL_IMAGE:figures/full_fig_p064_10.png]

**Figure 11.** Figure 11: Projection intervals on ℓ. In order for our construction to work, the projection interval Iγ for γ must be sufficiently large. We will require Tγ ⩾ T1 + LΛ. Equivalently, Tγ = log 1 θ ⩾ T0 + log 1 αΛ + LΛ, which is satisfied as long as 66 [PITH_FULL_IMAGE:figures/ful…

**Figure 12.** Figure 12: Small angles give long transverse rectangles. [PITH_FULL_IMAGE:figures/full_fig_p067_12.png]

**Figure 13.** Figure 13: A truncated rectangle at optimal height. [PITH_FULL_IMAGE:figures/full_fig_p070_13.png]

**Figure 14.** Figure 14: One side of the (β+, β− + )-maximal rectangle Rβ from [PITH_FULL_IMAGE:figures/full_fig_p074_14.png]

**Figure 15.** Figure 15: We will estimate the change in the height function along γ([a, b]). ℓ ℓ(a 0 ) α 0 − ℓ(b 0 ) β 0 − γ γ(t) = ℓ(0) θ ∼ log 1 θΛ γ(a) γ(b) p q [PITH_FULL_IMAGE:figures/full_fig_p075_15.png]

**Figure 16.** Figure 16: A corner segment with large angles. 79 [PITH_FULL_IMAGE:figures/full_fig_p079_16.png]

**Figure 17.** Figure 17: Outermost transverse rectangles determined by [PITH_FULL_IMAGE:figures/full_fig_p081_17.png]

**Figure 18.** Figure 18: An intersection point close to a negative value of the height function. [PITH_FULL_IMAGE:figures/full_fig_p082_18.png]

**Figure 19.** Figure 19: Quasigeodesics in two models for H2 . Definition 137. Suppose that I ⊆ R is a (possibly infinite) subinterval of R. Suppose that a: I → R+ is a function. We call the path α in the upper half space model of H2 given by α(t) = (t, a(t)) the path determined by the functi…

**Figure 20.** Figure 20: The radius functions for truncated projection intervals with close endpoints. [PITH_FULL_IMAGE:figures/full_fig_p086_20.png]

**Figure 21.** Figure 21: A non-exceptional geodesic intersecting an ideal polygon. [PITH_FULL_IMAGE:figures/full_fig_p087_21.png]

**Figure 22.** Figure 22: Radius functions for leaves with overlapping projection intervals. [PITH_FULL_IMAGE:figures/full_fig_p088_22.png]

**Figure 23.** Figure 23: The geodesic γ and the vertical flow lines. We denote by Fi the vertical flow segment from F0(ι(γ(ti))) to τγ(ti) = Fhγ (ti)(ι(γ(ti))). The path αi consisting of the union of the two geodesics [pi , qi ] ∪ [qi , ι(γ(ti))] is close to being a geodesic. In particular, 8…

**Figure 24.** Figure 24: The geodesic γ3 connecting non-adjacent limit points of γ1 and γ2. The four points γ1(0), γ2(0), γ1(t) and p determine a hyperbolic quadrilateral with three right angles, which is known is as a Lambert quadrilateral, and its sides satisfy sinh d = cosh tsinh θ, (36) s…

**Figure 25.** Figure 25: The geodesics of γ1 and γ2 intersect at angle θ. Using the sine formula for right angled triangles in hyperbolic space gives sin θ = sinh d sinh t . Using the elementary estimates: 1 2 θ ⩽ sin θ ⩽ θ for 0 ⩽ θ ⩽ 1 and 1 2 (e t − 1) ⩽ sinh t ⩽ 1 2 e t for t ⩾ 0, we get …

**Figure 26.** Figure 26: The nearest point projection from one leaf to an intersecting leaf. [PITH_FULL_IMAGE:figures/full_fig_p093_26.png]

**Figure 27.** Figure 27: A lower bound for the distance from γ1(t) to γ2. Applying the triangle inequality to the path from γ1(0) to γ1(t) via γ2(0) and γ2(r), gives t ⩽ θ + r + s. Similarly, applying the triangle inequality to the path from γ1(t) to γ2(t) via γ2(r), gives d ⩽ s + t − r. Addi…

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Determining subgroups via stationary measures
math.GT 2025-12 conditional novelty 8.0 of 10

Non-singular stationary measures for random walks on a discrete isometry group force the subgroups generated by the random walks to be commensurable.
Singularity of Cannon-Thurston maps
math.GT 2026-07 accept novelty 6.5 of 10

Pushforwards of full surface measures under Cannon-Thurston maps are mutually singular to 3-manifold measures, via distinct asymptotic time spent by typical geodesics near the base fiber.

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