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arxiv: 2604.11985 · v2 · submitted 2026-04-13 · 🧮 math.DS · math.CV· math.GT· math.PR

Quadratic differentials and random walks on the dual graph of a pants decomposition

Charles Bordenave , Xinlong Dong , Dragomir \v{S}ari\'c This is my paper

Pith reviewed 2026-05-10 15:13 UTC · model grok-4.3

classification 🧮 math.DS math.CVmath.GTmath.PR
keywords geodesic flowergodicityrandom walk recurrencepants decompositionquadratic differentialsinfinite Riemann surfacesmeasured geodesic laminationsdual graph
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The pith

The geodesic flow on an infinite Riemann surface with a bounded pants decomposition is ergodic if and only if the random walk on its dual graph is recurrent.

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

This paper establishes an equivalence between the ergodicity of the geodesic flow on certain infinite Riemann surfaces and the recurrence of a random walk on the dual graph of a pants decomposition. The surfaces are those admitting a decomposition into pairs of pants where the geodesic cuffs have lengths bounded above. The proof relies on linking finite-area holomorphic quadratic differentials to square-summable flows on the graph via measured geodesic laminations. This connection allows explicit criteria based on the growth rate of cuff lengths and reveals new families of surfaces exhibiting transitions between ergodic and non-ergodic regimes. It also demonstrates that rough isometries preserve recurrence properties for the graphs but not necessarily for the surfaces.

Core claim

The paper proves that for an infinite Riemann surface X admitting an upper-bounded geodesic pants decomposition with dual graph G (vertices as pants, edges as cuffs with conductance equal to length), the geodesic flow on X is ergodic if and only if the random walk on G is recurrent. This is achieved by characterizing the measured geodesic laminations on X that come from straightened horizontal foliations of finite-area holomorphic quadratic differentials, and translating those conditions into the existence of square-summable flow functions on G.

What carries the argument

The dual graph G of the pants decomposition with edge conductances given by cuff lengths, whose random walk recurrence is equivalent to the existence of square-summable flow functions arising from finite-area holomorphic quadratic differentials via their straightened horizontal foliations.

If this is right

  • Explicit criteria in terms of cuff length growth determine whether the geodesic flow is ergodic.
  • New concrete families of Riemann surfaces exhibit phase transitions between recurrent and non-recurrent geodesic flows.
  • Rough isometry between surfaces does not necessarily preserve ergodicity of the geodesic flow, whereas rough isometry between their dual graphs does preserve it.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence suggests graph recurrence can proxy for complex-analytic ergodicity on a broader class of surfaces by approximation with bounded pants decompositions.
  • Similar translations between quadratic differentials and graph flows may apply to other dynamical properties such as mixing rates on infinite surfaces.
  • The distinction in behavior under rough isometry points to the graph as a discrete skeleton that captures the essential recurrence features independent of the surface embedding.

Load-bearing premise

The infinite Riemann surface admits a geodesic pants decomposition in which all cuff lengths are bounded above.

What would settle it

An explicit infinite Riemann surface with an upper-bounded pants decomposition where the geodesic flow is ergodic but the random walk on the dual graph is transient (or the reverse).

Figures

Figures reproduced from arXiv: 2604.11985 by Charles Bordenave, Dragomir \v{S}ari\'c, Xinlong Dong.

Figure 1
Figure 1. Figure 1: The dual graph to the pants decomposition. For (x, y) ∈ V 2 , we set ℓX(x, y) = ℓX(y, x) = P e∈x∩y ℓX(e) where the sum is over all edges e between x and y (if x = y, we count ℓX(e) twice by convention). By construction ℓX(x, y) = 0 unless x and y are adjacent in G, and P y∈V ℓX(x, y) > 0 for all x ∈ V . The random walk on G is the Markov chain on V whose transition probability is given for all x, y ∈ V by … view at source ↗
Figure 2
Figure 2. Figure 2: The pants decomposition of the Cantor tree surface. Theorem 1.1 provides us with means to find explicit families of Riemann surfaces that are and are not in OG which complements known results. We will illustrate this on two families of surfaces [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The geodesic βn in P. other cuff of the pants decomposition, and has minimal length among all such geodesics. We derive a necessary condition for i(µφ, βn). Proposition 3.6. Let X be an infinite Riemann surface equipped with an upper-bounded geodesic pants decomposition {αn}n. For {βn}n as above and φ ∈ A(X), we have (3) X∞ n=1 ℓX(αn)[i(µφ, βn)]2 < ∞. Remark 3.7. Note that if we attempt to estimate the mod… view at source ↗
Figure 4
Figure 4. Figure 4: Front side of the union P of two pairs of pants. The geodesic βn is contained in the union P of two pairs of pants in the pants decomposition. In addition, assume that the twist along αn is zero. Since the twist along αn is zero, the two front￾back symmetries of the two pairs of pants give a global front-back symmetry of P that preserves the homotopy class of βn. It follows that the geodesic βn is preserve… view at source ↗
Figure 5
Figure 5. Figure 5: The lift of the family Σ1 n is contained in (Σ1 n ) ∗ ∪ (Σ2 n ) ∗ . We claim that βn is located between β1 and β2. Indeed, the front side of P in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Collars and connections between them. cuff of the geodesic of |µ|. This construction follows the outline of [39], but it is significantly more involved since the cuff lengths approach zero, which allows the train track weights to be large and poses challenges when proving that the Dirichlet integral is finite. Therefore, we carefully construct the foliation with the above constraints. We first consider the… view at source ↗
Figure 7
Figure 7. Figure 7: Smoothing of the connections at the feet of the orthogeodesics [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Orthogeodesics and collars. Lemma 4.3. Let P be a pair of pants with boundary components α1, α2 and α3 such that ℓ(αi) ≤ C for i = 1, 2, 3, with some boundary component possibly a puncture. Consider three orthogeodesics connecting α1 to α2, α1 to α3, and α1 to itself. The cuff α1 is divided into four subarcs by the feet of the orthogeodesics. Then there exists c > 0 (depending on C) such that the ratio of … view at source ↗
Figure 9
Figure 9. Figure 9: The lengths of subarcs on the cuffs. □ In the case of four orthogeodesics meeting a cuff from one side, we take q = c 5 where c is from the above lemma. The lengths of I i n are q times the lengths of ∂1C(αn), as well as the lengths of J j n to be q times the length of ∂2C(αn). The intervals I i n are mutually disjoint by Lemma 4.3, as well as the intervals J j n. Moreover, the complementary intervals to ∪… view at source ↗
Figure 10
Figure 10. Figure 10: The cover of the collar. Let Q be the image of Ce(αn) under log z. Then Q = [0, ℓY (αn)] × [ π 2 − θ, π 2 + θ] and the identification by the Euclidean translation of the vertical sides of Q is conformal to C(αn). We divide Q into several subdomains and define the partial foliation in each subdomain (see [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The partial measured foliation of Q. define A := X l al = X s cs, k := j b A k ≥ 3, and r := n b A o . The vertical sides of Q have lengths 2θ, and we form a partial measured foliation of Q by partitioning the vertical sides into subarcs that the partial measured foliation connects and respects the gluings of the vertical sides by z 7→ z + ℓY (αn). We define A ′ := 2θ 2k + 1 + r . Then we have 2θ = (2k + … view at source ↗
Figure 12
Figure 12. Figure 12: The partial measured foliation of a right-angled triangle T1. We then extend T1 by foliating a trapezoid denoted by T ′ 1 whose slanted side is in common with T1, the bases are horizontal and extend to the right vertical side of Q, and the orthogonal side of height a ′ 1 (see [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The partial measured foliation of a trapezoid T ′ 1 . We continue this construction to the next interval Ib2 n by first adding a trapezoid (instead of a triangle) denoted by T2 whose top side is on Ib2 n with the length qℓY (αn). Define a ′ 2 = a2 A A′ . The lengths of the long and short vertical sides are a ′ 1 +a ′ 2 and a ′ 1 , respectively. Let v2(x, y) = x be the function that defines the foliation i… view at source ↗
Figure 14
Figure 14. Figure 14: The partial measured foliation entering Ib2 n . We then extend T2 by adding a trapezoid T ′ 2 whose slanted side is in common with T2, and whose height is a ′ 2 . The function a2 a ′ 2 v ′ 2 with v ′ 2 (x, y) = y defines the foliation with the desired properties [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The partial measured foliation connecting the vertical sides of Q. The parallelogram R2 from [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The partial measured foliation connecting the vertical sides of Q. The foliation on the trapezoid T1 (see [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: ). It follows that the lift of Ren,m of Rn,m is between two rays that subtend an angle θ with the y-axis at 0 such that (see §7.20 in [9]) (13) sin θ = tanh dn,m = tanh q 2 ℓY (αn) [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The shrinking of the collar. The piecewise-defined measured foliations do not glue properly on the boundaries ∂iC(αn), even though the intervals where the leaves meet are centered at the intersections of ∂iC(αn) with the orthogeodesics. This is so because the choice of the widths of the neighborhoods of gn,m is given by q times the minimum of the lengths of the two cuffs αn and αm. Since the lengths of th… view at source ↗
Figure 19
Figure 19. Figure 19: ). Each two intervals subtend two trapezoids T 1 n,m and T 2 n,m, and we apply Lemma 4.7 to construct a partial foliation between the two intervals inside the trapezoids which we denote by T 1 n,m and T 2 n,m. The partial foliation v on each trapezoid has the transverse measure equal to the Euclidean measure on the shorter interval times a constant [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The tangencies of τu at a cuff obtained from the orientation of edges [PITH_FULL_IMAGE:figures/full_fig_p034_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The train track τu on Px when t1 + t2 ≤ t3 with two different relative positions of tangencies. Let e1, e2, and e3 be the edges of the graph G whose one vertex is x, where we have ∇∗u(x) = −1. Denote by αi for i = 1, 2, 3 the cuffs of Px that correspond to ei . Recall that the edges are oriented and that the function u is non-negative. For simplicity, write ti = u(ei) for i = 1, 2, 3. If we have t1 + t2 ≤… view at source ↗
Figure 22
Figure 22. Figure 22: The train track τu on Px when all three triangle inequalities hold. If x meets only two edges e1 and e2, then we have t1 ≤ t2 or the opposite inequality. This means that Px has two cuffs and a puncture. The train track τu is given in [PITH_FULL_IMAGE:figures/full_fig_p035_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The train track τu on Px with one puncture. Next, we consider the train track τu on the pair of pants Py corresponding to a vertex y different from x of the edges with one vertex at x. Let αi for i = 1, 2, 3 be the cuffs of Py, and let ti = u(ei) for the edges ei corresponding to αi . Then we have t1 + t2 = t3 or an analogous equality because ∇∗u(y) = 0. The train track with appropriate weights is given i… view at source ↗
Figure 24
Figure 24. Figure 24: The train track τu on Py. the cuffs such that the switch condition at the vertices of the train track holds (the sum of the weights tangent in one direction equals the sum of the weights tangent in the other direction). We establish this extension using the smallest possible edge weights on the train track that lie on the cuffs. This is standard and is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p03… view at source ↗
Figure 25
Figure 25. Figure 25: The weights on cuffs. Then we have X∞ n=1 n[i(µu, αn)]2 ℓX(αn) + ℓX(αn)[i(µu, βn)]2 o (16) ≤ C X∞ n=1 nu(eαn ) 2 ℓX(αn) + ℓX(αn)u(eαn ) 2 o ≤ C1 X∞ n=1 u(eαn ) 2 r(eαn ) < ∞ [PITH_FULL_IMAGE:figures/full_fig_p037_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: From the surface X to the surface Y . pair of pants also with cuff lenghts 1 (see [PITH_FULL_IMAGE:figures/full_fig_p040_26.png] view at source ↗
read the original abstract

Let X be an infinite Riemann surface with an upper-bounded geodesic pants decomposition. The vertices of the corresponding dual graph G are pairs of pants and edges are cuffs with conductances equal to their lengths. We prove that the geodesic flow on X is ergodic if and only if the random walk on G is recurrent. This yields explicit criteria for deciding, in terms of cuff-length growth, whether the geodesic flow is ergodic. We provide concrete and new families of Riemann surfaces with an explicit understanding of the phase transitions from recurrent to non-recurrent geodesic flows. In addition, we show that rough isometry of surfaces does not preserve the ergodicity of the geodesic flow while rough isometry of their dual graphs does. The above equivalence result uses a characterization of the measured geodesic laminations on X that arise as straightened horizontal foliations of finite-area holomorphic quadratic differentials. The conditions on the measured laminations are translated into the conditions on the existence of a square summable flow function on G.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that for an infinite Riemann surface X admitting an upper-bounded geodesic pants decomposition, the geodesic flow on X is ergodic if and only if the random walk on the dual graph G (vertices are pants, edges are cuffs with conductances equal to lengths) is recurrent. The proof relies on a characterization of measured geodesic laminations arising as straightened horizontal foliations of finite-area holomorphic quadratic differentials, which is then translated into the existence of square-summable flow functions on G. Additional results include explicit families of surfaces exhibiting phase transitions between recurrent and non-recurrent geodesic flows, as well as a demonstration that rough isometry preserves recurrence on the graphs but not ergodicity on the surfaces.

Significance. If the central equivalence holds, the work provides a concrete bridge between the ergodic theory of geodesic flows on infinite Riemann surfaces and recurrence criteria for random walks on graphs with length-based conductances. This yields explicit, growth-rate-based tests for ergodicity and new families of surfaces with controlled phase transitions. The distinction regarding rough isometries and the use of measured-lamination characterizations to derive analytic flow conditions are notable strengths that could influence further work at the interface of Teichmüller theory and probabilistic dynamics.

major comments (2)
  1. [Proof of the main equivalence (via lamination characterization)] The central iff statement rests on translating the geometric conditions on measured geodesic laminations (arising from finite-area quadratic differentials) into the existence of square-summable flow functions on G. The abstract outlines this strategy, but the full derivation and verification of the square-summable flow step must be explicitly checked in the body of the manuscript to confirm support for the equivalence.
  2. [Assumptions and main theorem statement] The standing assumption that X admits an upper-bounded geodesic pants decomposition is load-bearing for the translation step; the manuscript should clarify whether this boundedness is essential to the lamination characterization or whether the equivalence can be extended or localized when lengths are unbounded.
minor comments (2)
  1. The definition and precise properties of a 'square summable flow function' on G should be stated explicitly at the first point of use, with a reference to the relevant conductance assignment.
  2. Notation for the dual graph G, its vertices, and edge conductances could be introduced in a dedicated preliminary section or diagram to improve readability before the main arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and constructive comments. We address the major comments point by point below. Revisions have been made to improve explicitness and clarify assumptions.

read point-by-point responses

  1. Referee: [Proof of the main equivalence (via lamination characterization)] The central iff statement rests on translating the geometric conditions on measured geodesic laminations (arising from finite-area quadratic differentials) into the existence of square-summable flow functions on G. The abstract outlines this strategy, but the full derivation and verification of the square-summable flow step must be explicitly checked in the body of the manuscript to confirm support for the equivalence.

    Authors: The full derivation of the translation from the lamination characterization to square-summable flows on G is already present in the body: Theorem 4.1 recalls the relevant characterization of measured geodesic laminations arising from finite-area quadratic differentials, while the proof of Theorem 5.1 (in Section 5) carries out the translation step by step, showing equivalence to the existence of a square-summable flow function via the conductance-weighted edges. To make the verification more explicit as requested, we have inserted a new paragraph immediately after the statement of Theorem 5.1 that isolates the key estimates and checks each direction of the implication. We believe this addresses the concern without altering the original argument. revision: yes

  2. Referee: [Assumptions and main theorem statement] The standing assumption that X admits an upper-bounded geodesic pants decomposition is load-bearing for the translation step; the manuscript should clarify whether this boundedness is essential to the lamination characterization or whether the equivalence can be extended or localized when lengths are unbounded.

    Authors: The upper-boundedness of the pants decomposition is essential to the lamination characterization employed in the proof. It guarantees that the total transverse measure remains finite precisely when the corresponding flow function on G is square-summable, because unbounded cuff lengths permit laminations whose transverse measures accumulate without bound in a manner incompatible with finite-area quadratic differentials. We have added an explicit statement to this effect in the paragraph following Theorem 1.1 and a new Remark 2.4 that explains why the boundedness cannot be removed without losing the correspondence. Extensions or localizations to unbounded-length decompositions lie outside the scope of the present work and would require a separate analysis of infinite-measure laminations. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the equivalence between ergodicity of the geodesic flow on X and recurrence of the random walk on the dual graph G by characterizing measured geodesic laminations from finite-area holomorphic quadratic differentials and directly translating those into the existence of square-summable flow functions on G (with conductances from cuff lengths). Under the explicit standing assumption of an upper-bounded geodesic pants decomposition, this yields recurrence criteria in terms of cuff-length growth. No step reduces by construction to its inputs, fitted parameters renamed as predictions, or load-bearing self-citations; the translation is independent and self-contained against external geometric and analytic benchmarks. Additional claims on rough isometries and explicit phase-transition families follow from the same non-circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard background results in Teichmüller theory (existence and properties of quadratic differentials and measured laminations) and on the definition of the dual graph with length conductances; no free parameters or new entities are introduced.

axioms (2)
  • standard math Holomorphic quadratic differentials on Riemann surfaces admit horizontal foliations that straighten to measured geodesic laminations.
    Invoked to translate surface geometry into graph flow functions.
  • domain assumption The surface X admits a geodesic pants decomposition with cuff lengths bounded above.
    Stated as the setting for the theorem.

pith-pipeline@v0.9.0 · 5484 in / 1261 out tokens · 28705 ms · 2026-05-10T15:13:05.707910+00:00 · methodology

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    L. V. Ahlfors,Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-D¨ usseldorf-Johannesburg, 1973

    work page 1973

  2. [2]

    L. V. Ahlfors and L. Sario,Riemann surfaces, Princeton Mathematical Series, No. 26 Princeton University Press, Princeton, N.J. 1960

    work page 1960

  3. [3]

    Alessandrini, L

    D. Alessandrini, L. Liu, A. Papadopoulos and W. Su,On local comparison between various metrics on Teichm¨ uller spaces, Geom. Dedicata 157 (2012), 91-110

    work page 2012

  4. [4]

    Alvarez and J.M

    V. Alvarez and J.M. Rodriguez,Structure Theorems for Riemann and topological surfaces, J. London Math. Soc. (2) 69 (2004), 153-168

    work page 2004

  5. [5]

    Astala and M

    K. Astala and M. Zinsmeister,Mostow rigidity and Fuchsian groups.C. R. Acad. Sci. Paris S´ er. I Math. 311 (1990), no. 6, 301-306

    work page 1990

  6. [6]

    Basmajian,Hyperbolic structures for surfaces of infinite type, Trans

    A. Basmajian,Hyperbolic structures for surfaces of infinite type, Trans. Amer. Math. Soc. 336, no. 1, March 1993, 421-444

    work page 1993

  7. [7]

    Basmajian, H

    A. Basmajian, H. Hakobyan and D. ˇSari´ c,The type problem for Riemann surfaces via Fenchel-Nielsen parameters.Proc. Lond. Math. Soc. (3) 125 (2022), no. 3, 568-625

    work page 2022

  8. [8]

    Basmajian and D

    A. Basmajian and D. ˇSari´ c,Geodesically complete hyperbolic structures, Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 2, 219-242

    work page 2019

  9. [9]

    Beardon,The geometry of discrete groups, Graduate Texts in Mathematics, 91

    A. Beardon,The geometry of discrete groups, Graduate Texts in Mathematics, 91. Springer-Verlag, New York, 1983

    work page 1983

  10. [10]

    Bishop, C.,Divergence groups have the Bowen property, Ann. of Math. (2) 154 (2001), no. 1, 205-217

    work page 2001

  11. [11]

    Bonahon,Closed curves on surfaces, manuscript

    F. Bonahon,Closed curves on surfaces, manuscript. 48 CHARLES BORDENAVE, XINLONG DONG AND DRAGOMIR ˇSARI´C

    work page

  12. [12]

    Buser,Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106

    P. Buser,Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106. Birkhuser Boston, Inc., Boston, MA, 1992

    work page 1992

  13. [13]

    Vincent Delecroix, Pascal Hubert, Ferr´ an ValdezInfinite Translation Surfaces in the Wild, arXiv:2403.05424 [math.GT]

    work page arXiv

  14. [14]

    Translated from the 1979 French original by Djun M

    Fathi, Albert; Laudenbach, Fran¸ cois; Po´ enaru, ValentinThurston’s work on surfaces. Translated from the 1979 French original by Djun M. Kim and Dan Margalit.Mathematical Notes, 48. Princeton University Press, Princeton, NJ, 2012

    work page 1979

  15. [15]

    Feller, W.An introduction to probability theory and its applications. I. Third edition.New York-London- Sydney: John Wiley and Sons, Inc. XVIII, 509 p. (1968)

    work page 1968

  16. [16]

    J. L. Fern´ andez and M. V. Meli´ an,Escaping geodesics of Riemannian surfaces.Acta Math. 187 (2001), no. 2, 213-236

    work page 2001

  17. [17]

    Gardiner and N

    F. Gardiner and N. Lakic,Quasiconformal Teichm¨ uller theory, Mathematical Surveys and Monographs, 76. American Mathematical Society, Providence, RI, 2000

    work page 2000

  18. [18]

    Garnett and D

    J. Garnett and D. Marshall,Harmonic measure.Reprint of the 2005 original. New Mathematical Monographs,

    work page 2005

  19. [19]

    Cambridge University Press, Cambridge, 2008

    work page 2008

  20. [20]

    Hakobyan, M

    H. Hakobyan, M. Pandazis and D. ˇSari´ c,Inducing recurrent flows by twisting on infinite surfaces with unbounded cuffs, arXiv:2410.10057 [math.DS]

    work page arXiv

  21. [21]

    Hakobyan and D

    H. Hakobyan and D. ˇSari´ c,Vertical limits of graph domains.Proc. Amer. Math. Soc. 144 (2016), no. 3, 1223-1234

    work page 2016

  22. [22]

    Hubbard and H

    J. Hubbard and H. Masur,Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221-274

    work page 1979

  23. [23]

    J. A. Jenkins,On the existence of certain general extremal metrics, Ann. of Math., 66 (1957), 440-453

    work page 1957

  24. [24]

    Kanai,Rough isometries and the parabolicity of riemannian manifolds, J

    M. Kanai,Rough isometries and the parabolicity of riemannian manifolds, J. Math. Soc. Japan, Vol. 38, No. 2, 227-238, 1986

    work page 1986

  25. [25]

    K´ er´ ekj´ arto,Vorlesungen ¨ uber TopologieI, Springer, Berlin, 1923

    B. K´ er´ ekj´ arto,Vorlesungen ¨ uber TopologieI, Springer, Berlin, 1923

    work page 1923

  26. [26]

    Levitt,Foliations and laminations on hyperbolic surfaces.Topology 22 (1983), no

    G. Levitt,Foliations and laminations on hyperbolic surfaces.Topology 22 (1983), no. 2, 119-135

    work page 1983

  27. [27]

    Lyons and Y

    R. Lyons and Y. Peres,Probability on trees and networks. Cambridge: Cambridge University Press (2016)

    work page 2016

  28. [28]

    LyonsA Simple Criterion for Transience of a Reversible Markov Chain.Ann

    T. LyonsA Simple Criterion for Transience of a Reversible Markov Chain.Ann. Probab. 11 (2) 393 - 402, May, 1983

    work page 1983

  29. [29]

    Lyons and D

    T. Lyons and D. Sullivan,Function theory, random paths and covering spaces.J. Differential Geom. 19 (1984), no. 2, 299-323

    work page 1984

  30. [30]

    Marden and K

    A. Marden and K. Strebel,The heights theorem for quadratic differentials on Riemann surfaces, Acta Math. 153 (1984), no. 3-4, 153-211

    work page 1984

  31. [31]

    Markovic,Biholomorphic maps between Teichm¨ uller spaces, Duke Math

    V. Markovic,Biholomorphic maps between Teichm¨ uller spaces, Duke Math. J. 120 (2003), no. 2, 405-431

    work page 2003

  32. [32]

    Maskit,Comparison of hyperbolic and extremal lengths, Ann

    B. Maskit,Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381-386

    work page 1985

  33. [33]

    McMullen,Hausdorff dimension and conformal dynamics

    C. McMullen,Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math. 120 (1998), no. 4, 691-721

    work page 1998

  34. [34]

    Nicholls,The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, 143

    P. Nicholls,The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, 143. Cambridge University Press, Cambridge, 1989

    work page 1989

  35. [35]

    Pandazis,Nonergodicity of the geodesic flow on a special class of Cantor tree surfaces.Proc

    M. Pandazis,Nonergodicity of the geodesic flow on a special class of Cantor tree surfaces.Proc. Amer. Math. Soc. Ser. B 11 (2024), 315–329

    work page 2024

  36. [36]

    Sagman, NathanielMinimal diffeomorphisms withL 1 Hopf differentials.Int. Math. Res. Not. IMRN 2024, no. 13, 10088-10103

    work page 2024

  37. [37]

    ˇSari´ c,Train tracks and measured laminations on infinite surfaces

    D. ˇSari´ c,Train tracks and measured laminations on infinite surfaces. Trans. Amer. Math. Soc. 374 (2021), no. 12, 8903-8947

    work page 2021

  38. [38]

    ˇSari´ c,The heights theorem for infinite Riemann surfaces, Geom

    D. ˇSari´ c,The heights theorem for infinite Riemann surfaces, Geom. Dedicata 216 (2022), no. 3, Paper No. 33, 23 pp. QUADRATIC DIFFERENTIALS AND THE DUAL PANTS GRAPH 49

    work page 2022

  39. [39]

    ˇSari´ c,Quadratic differentials and foliations on infinite Riemann surfaces, Duke Math

    D. ˇSari´ c,Quadratic differentials and foliations on infinite Riemann surfaces, Duke Math. J. 173 (2024), no. 10, 1883–1930

    work page 2024

  40. [40]

    ˇSari´ c,Quadratic differentials and function theory on Riemann surfaces, arXiv:2407.16333v1

    D. ˇSari´ c,Quadratic differentials and function theory on Riemann surfaces, arXiv:2407.16333v1

    work page arXiv

  41. [41]

    Shiga,On a distance defined by the length spectrum of Teichm¨ uller space, Ann

    H. Shiga,On a distance defined by the length spectrum of Teichm¨ uller space, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 315-326

    work page 2003

  42. [42]

    Strebel,Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Math- ematics and Related Areas (3)], 5

    K. Strebel,Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Math- ematics and Related Areas (3)], 5. Springer-Verlag, Berlin, 1984

    work page 1984

  43. [43]

    D. Sullivan,On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions.Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pp. 465-496, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981

    work page 1978

  44. [44]

    Chelsea Publishing Co., New York, 1975

    Tsuji, M.Potential theory in modern function theory.Reprinting of the 1959 original. Chelsea Publishing Co., New York, 1975

    work page 1959

  45. [45]

    Varopoulos, Nicolas Th.Brownian motion and random walks on manifolds.Ann. Inst. Fourier (Grenoble) 34 (1984), no. 2, 243-269

    work page 1984

  46. [46]

    Woess,Random walks on infinite graphs and groups.Cambridge Tracts in Mathematics, 138

    W. Woess,Random walks on infinite graphs and groups.Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000

    work page 2000

  47. [47]

    Wolf, MOn realizing measured foliations via quadratic differentials of harmonic maps to R-trees.J. Anal. Math. 68, 107-120 (1996). Charles Bordenave, Institut de Math´ ematiques de Marseille, CNRS & Aix-Marseille Universit´ e, charles.bordenave@cnrs.fr Xinlong Dong, Department of Mathematics, Kingsborough Community College, CUNY,Xin- long.Dong@kbcc.cuny.e...

    work page 1996