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REVIEW 4 major objections 5 minor 102 references

The geometry of totally geodesic subvarieties of moduli spaces of Riemann surfaces

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Boundary of every geodesic subvariety splits into diagonal factors.

desk verdict The semisimplicity theorem is the real result and holds up; the HHS/HHG part is genuinely incomplete as written and needs referee pressure. read the letter →

arxiv 2412.16330 v2 pith:3Y5NWASF submitted 2024-12-20 math.GT math.DSmath.GR

Francisco Arana-Herrera , Alex Wright This is my paper · ORCID
classification math.GTmath.DSmath.GR MSC 32G1530F6014H1520F65
keywords totallygeodesicsubmanifoldsTeichmüllerspaceDeligne–Mumfordcompactificationsemisimpleboundaryhierarchicallyhyperbolicspacesquadraticdifferentialsinvariantsubvarietiesmoduli
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

The paper proves that every algebraic totally geodesic submanifold of Teichmüller space meets the Deligne–Mumford bordification in a semisimple way: the intersection of its closure with any boundary stratum is a product of simple factors, each of which sits in its product of Teichmüller spaces like a diagonal embedding. It then shows that this boundary structure forces the submanifold itself, and the orbifold fundamental group of its image in moduli space, to be hierarchically hyperbolic. This matters because it confirms that higher-dimensional totally geodesic submanifolds behave like whole Teichmüller spaces, and it gives a structural handle on their boundary that can be used in classification problems.

What carries the argument

The decisive object is the semisimple decomposition of the boundary locus L, obtained by passing to the associated invariant subvariety of quadratic differentials and slicing it so that all forgotten components carry zero differentials. The engine of the decomposition is a previously established structure theorem, quoted as Theorem 4.5, stating that in any prime invariant subvariety of a product of strata of quadratic differentials the absolute periods in one component locally determine the absolute periods in every other component; this turns the relevant variety into a product of primes. A non-standard exponential map, defined by flowing along the differential subspaces corresponding to each prime factor, is shown to be continuous, injective, proper, and surjective onto L, which reveals L as a product of simple factors. A separate smoothness argument, using Lipschitz regularity of complex analytic sets, upgrades L from a complex analytic set to a complex submanifold.

What would settle it

Find an algebraic totally geodesic submanifold N and a stratum of the Deligne–Mumford bordification for which an irreducible component of the intersection of N with that stratum is not a product of simple factors, for instance a component containing two Teichmüller coordinates whose distance ratios are not equal. The low-genus examples can be checked computationally: if any of their boundary components fails to decompose as a diagonal product, Theorem 1.3 is false.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: if N is an algebraic totally geodesic submanifold of Teichmüller space and L is the intersection of N with a stratum of the Deligne–Mumford bordification, then L is semisimple and algebraic. Semisimple means L is a product of simple factors, and a simple factor is one whose projection to each Teichmüller coordinate is an isometric embedding, so that distances in all coordinates agree and the factor looks metrically like a diagonal copy of a smaller Teichmüller space. The proof first shows that each boundary component is itself totally geodesic and algebraic, then uses invariance under the GL(2,R)-action on quadratic differentials to decompose the associated variety into prime factors, and finally constructs a homeomorphism from a sum of differential subspaces to L that forces the product structure. As a consequence, Theorem 1.5 states that N and the stabilizer of N in the mapping class group are hierarchically hyperbolic, meaning a carefully chosen collection of subsurfaces and their curve graphs serve as coarse coordinates for N.

Load-bearing premise

The load-bearing premise is an external structure theorem: in any irreducible non-product invariant subvariety of a product of strata of quadratic differentials, the absolute periods in one component locally determine the absolute periods in every other component; if this theorem were false, the semisimplicity of the boundary would not follow from the given argument.

Editorial extensions

If this is right

  • Every boundary component of an algebraic totally geodesic submanifold is a product of diagonally embedded simple factors, so the way the submanifold approaches the Deligne–Mumford boundary is extremely constrained.
  • The submanifold N and its orbifold fundamental group are hierarchically hyperbolic, so the Realization Theorem and Distance Formula hold for them with respect to a selected family of subsurface curve graphs.
  • Cylinder curves on the associated quadratic differential variety split into equivalence classes with constant ratios of circumferences and moduli; pinching exactly the unions of such equivalence classes produces the boundary strata of N.
  • The equivalence relation on subsurfaces gives both rigidity (equivalent subsurfaces share shape) and flexibility (independent subsurfaces vary independently), matching the paper's claimed new rigidity and new flexibility.
  • The boundary semisimplicity supplies an inductive tool for the classification of higher-dimensional totally geodesic subvarieties of moduli space.

Reading between the lines

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

  • If the semisimplicity theorem is correct, it suggests an inductive classification strategy: reconstruct a totally geodesic subvariety from its lower-dimensional boundary factors, each of which is itself a totally geodesic object of the same type.
  • The paper's equivalence relation on curves and subsurfaces is reminiscent of a root-system-like combinatorial data structure; one testable extension is whether this structure, together with a finite list of boundary factors, uniquely determines the original submanifold.
  • Hierarchical hyperbolicity implies standard consequences such as finite asymptotic dimension, classification of maximal quasiflats, and undistortedness of the fundamental group in the mapping class group; these follow from general theory once Theorem 1.5 is accepted, though the paper only notes some of them.
  • The boundary decomposition may extend to non-algebraic totally geodesic submanifolds if algebraicity turns out to be automatic from the definition; the paper's arguments, however, use algebraicity at several essential points.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proves two main theorems about algebraic totally geodesic submanifolds N of Teichmüller space. Theorem 1.3 states that the intersection L of the closure of N with any stratum of the Deligne–Mumford bordification is semisimple and algebraic, i.e. a product of simple factors that are metrically diagonal embeddings. The proof combines the Cylinder Deformation Theorem, results on GL(2,R)-invariant subvarieties of quadratic differentials, and the Chen–Wright primality theorem for products of strata. Theorem 1.5 states that N itself is a hierarchically hyperbolic space and that the stabilizer of N in the mapping class group is a hierarchically hyperbolic group. The proof of Theorem 1.5 is given in Section 9 by verifying the axioms of an almost HHS and then invoking an equivalence with HHS. The paper also states Theorem 1.6, a collection of structural properties of cylinder curves and equivalence classes for N.

Significance. If fully established, Theorem 1.3 is a major structural result: it gives a clean inductive description of the boundary of every algebraic totally geodesic submanifold, and the semisimplicity conclusion is already corroborated by independent simultaneous work of Benirschke–Dozier–Rached [BDR24]. The proof of Theorem 1.3 in Sections 2–5 is detailed and appears sound, and it makes novel use of coarse-geometric and dynamical input. Theorem 1.5 is also significant as a concrete realization of the authors' Metaconjecture 1.2, since it extends the hierarchical hyperbolicity of Teichmüller space and mapping class groups to these submanifolds and their fundamental groups. The paper is ambitious and contains many useful auxiliary results, especially Theorem 2.1, Lemma 2.13, and the electrification results of Appendix B. However, as written, Theorem 1.5 is not fully proved: Section 9 contains several explicit sketches and deferred verifications, including the container axiom for HHGs, so the second main theorem is not established at the same level of rigor as Theorem 1.3.

major comments (4)
  1. [Section 9.3] The hierarchically hyperbolic group structure for the stabilizer Γ_N is explicitly not proved in full: the text states 'We will only sketch this, leaving some details to the reader.' In particular, the partial realization axiom for annular domains is asserted without a complete argument, and the construction of the HHS on the Cayley graph depends on a co-compactness step in the thick part of N that is only mentioned. Since Theorem 1.5's second assertion is one of the two main theorems of the paper, this is a load-bearing gap. The authors should either supply the missing details or explicitly state which parts of Theorem 1.5 are conditional.
  2. [Section 9.4] The proof of Theorem 1.5 relies on the assertion that every almost HHS is an HHS, citing [ABD21, Appendix], but the text immediately notes a 'mild complication' for HHGs and refers to [ABR23, Remark 3.4]. The promised sketch of the usual orthogonality/container axiom is not actually supplied: the paragraph beginning 'If one uses a slightly larger set of domains' describes an idea but does not verify the container axiom for the constructed domains. Because the container axiom is part of the HHS definition and the cited equivalence is not uniform for HHGs, the paper does not yet establish Theorem 1.5 as written. A complete verification of the container axiom, or a precise citation of a theorem that covers the HHG case, is needed.
  3. [Section 9.2, Partial realization] For the partial realization axiom, the proof for annular domains is only a sketch: the text says 'we only provide a sketch' and then invokes Lemma 3.16 and the proof of Lemma 7.3 without showing that the unpinching operation preserves the previously realized non-annular projections while adjusting the annular twist coordinates. This is a necessary step for the coarse surjectivity of annular projections and for the subsequent HHG argument. The authors should give a complete proof or a precise reduction to the cited lemmas.
  4. [Section 9.1] The paper states that the axioms for an almost HHS are 'paraphrased imprecisely, omitting a number of details' and that the reader should consult other sources. For a theorem whose proof is the content of the section, this level of imprecision makes it difficult to verify that the constructed structure satisfies all axioms with uniform constants. In particular, the verifications of axioms (4c), (6), and (7) are asserted to 'follow from' the corresponding Teichmüller space axioms without a detailed argument. Since Theorem 1.5 is central, the authors should either use a precise axiomatic framework throughout or clearly identify which axioms are being verified and which are being imported.
minor comments (5)
  1. [Section 1.1] There is a typo: 'Koyabashi metric' should be 'Kobayashi metric'.
  2. [Section 9.3] The sentence 'This conclude our sketch' should be 'This concludes our sketch.'
  3. [Appendix A] The phrase 'Rafi has proven proven a no backtracking result' contains a duplicated word 'proven'.
  4. [Section 2.4] The sentence 'We think of the Teichmüller space version as mapping to' is grammatically incomplete; consider revising to 'We think of the Teichmüller space version as mapping to the set {x+iy : y ≥ 1} ⊂ H.'
  5. [Appendix B] In the proof of Theorem B.1, the notation 'EpIq' appears to be a typo for 'E'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the semisimplicity and hierarchical hyperbolicity theorems are derived from external prior results and independent geometric arguments, with no step reducing to its own input by construction.

full rationale

The paper's derivation chain was inspected for the defined circularity patterns. Theorem 1.3 is proved by first establishing that boundary intersections are totally geodesic subsets (Theorem 3.9, Section 3) and then, in Section 4.2, converting the associated GL(2,R)-invariant variety into a product of primes using Chen–Wright's Theorem 4.5. That theorem is a prior published result with an independent proof; the paper then uses slices, a non-standard exponential map, and invariance-of-domain arguments (Lemmas 4.6–4.10) to lift the prime decomposition into the geometric product-of-simple-factors conclusion. The conclusion is therefore not identical to the cited theorem by construction. Theorem 1.5's HHS proof builds domains from active subsurfaces and verifies the axioms using Theorem 1.3, Rafi's theory, and standard hierarchical hyperbolicity results; no fitted parameter is renamed as a prediction and no definition contains the target conclusion. The paper does lean on several prior results of the authors (e.g., [Wri20], [MW17], [CW21]), but these are independently proved theorems and are used as tools, not as a substitute for the argument. The central theorem also has independent simultaneous corroboration [BDR24]. Two limitations are explicitly flagged in the text: Section 9.3 says of the HHG structure 'We will only sketch this, leaving some details to the reader,' and Section 9.4 concedes 'there is a mild complication in the equivalence of hierarchically hyperbolicity and almost hierarchically hyperbolicity for HHGs [ABR23, Remark 3.4].' These are completeness gaps in the proof of Theorem 1.5, not cases of a claim reducing to its own input. Accordingly no circular step is found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data. The proof relies on a network of deep prior theorems in Teichmuller theory, dynamics, and group theory; none of the central results reduce to an input assumption by construction.

assumptions (5)
  • domain assumption Wright's theorem that every higher dimensional totally geodesic submanifold is algebraic and there are finitely many (Theorem 1.1).
    Used throughout to justify that the submanifold under consideration is algebraic. Cited from [Wri20], not reproved.
  • domain assumption Chen-Wright structure theorem for prime invariant subvarieties of products of strata: absolute periods in one component locally determine those in another (Theorem 4.5).
    Key input in Section 4.2 to decompose the invariant variety M into a product of primes, yielding the semisimple splitting.
  • domain assumption Rafi's active interval theory and formulas for annular projections, including no-backtracking and quasi-geodesic behavior (Theorem B.3, Appendix A).
    Used to control subsurface projections along Teichmuller geodesics in Sections 2, 7, 8, and Appendix B. The paper proves the auxiliary electrification theorem (Theorem B.1) using this theory.
  • domain assumption The equivalence between almost HHS and HHS, with the caveat noted in [ABR23, Remark 3.4].
    Used in Section 9 to verify the HHS axioms for a slight modification; the paper notes a complication and gives a sketch to address it in Section 9.4.
  • domain assumption Cylinder Deformation Theorem and properties of GL(2,R) invariant subvarieties of quadratic differentials, including algebraicity results.
    Used in Sections 2, 3, and 6 to produce and deform cylinders and to ensure algebraicity of projections.

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Pith. "Pith review of The geometry of totally geodesic subvarieties of moduli spaces of Riemann surfaces." pith.science (2026). https://pith.science/paper/3Y5NWASF

@misc{pith2026241216330,
  author       = {Pith},
  title        = {Pith review of: The geometry of totally geodesic subvarieties of moduli spaces of Riemann surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3Y5NWASF}},
  note         = {Machine review of arXiv:2412.16330}
}
read the original abstract

We prove a semisimplicity result for the boundary, in the corresponding Deligne-Mumford compactification, of a totally geodesic subvariety of a moduli space of Riemann surfaces. At the level of Teichm\"uller space, this semisimplicity theorem gives that each component of the boundary is a product of simple factors, each of which behaves metrically like a diagonal embedding. Building on this result, we also show that the associated totally geodesic submanifolds of Teichm\"uller space and orbifold fundamental groups are hierarchically hyperbolic. The proof intertwines in a novel way results and perspectives originating in dynamics, algebraic geometry, geometric group theory, and both classical and modern Teichm\"uller theory. It establishes both new rigidity and new flexibility for totally geodesic submanifolds and their associated varieties and orbifold fundamental groups and provides a rich set of new tools for the study of these objects.

Figures

Figures reproduced from arXiv: 2412.16330 by the authors.

Figure 1
Figure 1. The cover X Ñ Y in Example 1.4. Our second main result is the following, whose proof uses Theorem 1.3. Theorem 1.5. Let N Ď Tg,n be an algebraic totally geodesic submanifold of Te￾ichm¨uller space. Then N is a hierarchically hyperbolic space and the associated orbifold fundamental group is a hierarchically hyperbolic group. By “the associated orbifold fundamental group” we mean the stabilizer of N in the mapping cla… view at source ↗
Figure 2
Figure 2. Horocycle arcs joining pairs of points in Teichm¨uller space. Arrows represent positive time directions. Theorem 1.3], which says that QmaxN is the lift of a particular kind of invariant subvariety, more precisely, it has no rel, and then invoke [MW17, Corollary 1.6], which gives the desired result. (See also [AW24, Lemma 6.10] for a different proof of the result powering [MW17, Corollary 1.6].) □ We are now ready t… view at source ↗
Figure 3
Figure 3. Dotted lines are Teichm¨uller horocycles and solid straight lines are Teichm¨uller geodesics. Let F : r0, 1s Ñ N denote the reparametrized Teichm¨uller geodesic from X1 to Y 1 . Now, by Corollary 2.7, for every Y 2 P F we can find a continuously varying horizontally periodic quadratic differential on Y 2 whose horizontal cylinder core curves are exactly those corresponding to cylinders in rCs. Using the Cylinder Def… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: If N is the whole Teichm¨uller space, pinching the curves drawn gives a simple factor which is a point in T pU1q ˆ T pU2q for the trivial reason that T pU1q and T pU2q are points them￾selves. In this example each curve is only equivalent to itself. It remains only to a…
Figure 5
Figure 5. Figure 5: The path of Lemma A.3. A technical point is that the max in the formula for πα does not cause problems; a priori one might worry that significant change to the real part might take place during an interval when the imaginary part is constantly equal to 1. For this one …
Figure 6
Figure 6. Figure 6: The proof of Corollary B.2. We also have a 1-Lipschitz inclusion ι1 : E1 Ñ E, and π1 “ π ˝ ι1. The situation is summarized in [PITH_FULL_IMAGE:figures/full_fig_p047_6.png]

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Works this paper leans on

102 extracted references · 76 canonical work pages

  1. [1]

    Carolyn Abbott, Jason Behrstock, and Matthew Gentry Durham, Largest acylindrical actions and stability in hierarchically hyperbolic groups, Trans. Amer. Math. Soc. Ser. B 8 (2021), 66--104, With an appendix by Daniel Berlyne and Jacob Russell

    2021

  2. [2]

    Carolyn Abbott, Jason Behrstock, and Jacob Russell, Structure invariant properties of the hierarchically hyperbolic boundary, 2023, arXiv:2208.07930

    work page Pith review arXiv 2023

  3. [3]

    Differential Geom

    David Aulicino and Duc-Manh Nguyen, Rank 2 affine manifolds in genus 3, J. Differential Geom. 116 (2020), no. 2, 205--280

    2020

  4. [4]

    Antonakoudis, The complex geometry of T eichm\" u ller spaces and bounded symmetric domains , In the tradition of A hlfors- B ers

    Stergios M. Antonakoudis, The complex geometry of T eichm\" u ller spaces and bounded symmetric domains , In the tradition of A hlfors- B ers. VII , Contemp. Math., vol. 696, Amer. Math. Soc., Providence, RI, 2017, pp. 1--16

    2017

  5. [5]

    , Isometric disks are holomorphic, Invent. Math. 207 (2017), no. 3, 1289--1299

    2017

  6. [6]

    , Teichm\" u ller spaces and bounded symmetric domains do not mix isometrically , Geom. Funct. Anal. 27 (2017), no. 3, 453--465

    2017

  7. [7]

    Paul Apisa, GL _2 R orbit closures in hyperelliptic components of strata , Duke Math. J. 167 (2018), no. 4, 679--742

    2018

  8. [8]

    , Invariant S ubvarieties of M inimal H omological D imension, Z ero L yapunov E xponents, and M onodromy , Geom. Funct. Anal. 35 (2025), no. 1, 113--160

    2025

Show all 102 references
  1. [9]

    Taylor, Covers of surfaces, K leinian groups and the curve complex , J

    Tarik Aougab, Priyam Patel, and Samuel J. Taylor, Covers of surfaces, K leinian groups and the curve complex , J. Topol. 15 (2022), no. 4, 1833--1863

    2022

  2. [10]

    Paul Apisa and Alex Wright, Generalizations of the E ierlegende- W ollmilchsau , Camb. J. Math. 10 (2022), no. 4, 859--933

    2022

  3. [11]

    , High rank invariant subvarieties, Ann. of Math. (2) 198 (2023), no. 2, 657--726

    2023

  4. [12]

    , Reconstructing orbit closures from their boundaries, Mem. Amer. Math. Soc. 298 (2024), no. 1487, v+141

    2024

  5. [13]

    Matthew Charles Bainbridge, Euler characteristics of T eichmuller curves in genus two , ProQuest LLC, Ann Arbor, MI, 2006, Thesis (Ph.D.)--Harvard University

    2006

  6. [14]

    Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin M\" o ller, Compactification of strata of A belian differentials , Duke Math. J. 167 (2018), no. 12, 2347--2416

    2018

  7. [15]

    Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin M\"oller, The moduli space of multi-scale differentials, 2019, arXiv:1910.13492

    2019 arXiv

  8. [16]

    Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin M\" o ller, Strata of k -differentials , Algebr. Geom. 6 (2019), no. 2, 196--233

    2019

  9. [17]

    Frederik Benirschke, Benjamin Dozier, and Samuel Grushevsky, Equations of linear subvarieties of strata of differentials, Geom. Topol. 26 (2022), no. 6, 2773--2830

    2022

  10. [18]

    Frederik Benirschke, Benjamin Dozier, and John Rached, The boundary of a totally geodesic subvariety of moduli space, 2024, arXiv:2412.06765

    2024 arXiv

  11. [19]

    Frederik Benirschke, The boundary of linear subvarieties, J. Eur. Math. Soc. (JEMS) 25 (2023), no. 11, 4521--4582

    2023

  12. [20]

    Lipman Bers, On spaces of R iemann surfaces with nodes , Bull. Amer. Math. Soc. 80 (1974), 1219--1222

    1974

  13. [21]

    , Finite-dimensional T eichm\" u ller spaces and generalizations , Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 131--172

    1981

  14. [22]

    Birbrair, A

    L. Birbrair, A. Fernandes, D. T. L\^ e , and J. E. Sampaio, Lipschitz regular complex algebraic sets are smooth, Proc. Amer. Math. Soc. 144 (2016), no. 3, 983--987

    2016

  15. [23]

    Hagen, and Alessandro Sisto, Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups, Proc

    Jason Behrstock, Mark F. Hagen, and Alessandro Sisto, Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups, Proc. Lond. Math. Soc. (3) 114 (2017), no. 5, 890--926

    2017

  16. [24]

    , Hierarchically hyperbolic spaces, I : C urve complexes for cubical groups , Geom. Topol. 21 (2017), no. 3, 1731--1804

    2017

  17. [25]

    Jason Behrstock, Mark Hagen, and Alessandro Sisto, Hierarchically hyperbolic spaces II : C ombination theorems and the distance formula , Pacific J. Math. 299 (2019), no. 2, 257--338

    2019

  18. [26]

    Hagen, and Alessandro Sisto, Quasiflats in hierarchically hyperbolic spaces, Duke Math

    Jason Behrstock, Mark F. Hagen, and Alessandro Sisto, Quasiflats in hierarchically hyperbolic spaces, Duke Math. J. 170 (2021), no. 5, 909--996

    2021

  19. [27]

    Jason Behrstock, Bruce Kleiner, Yair Minsky, and Lee Mosher, Geometry and rigidity of mapping class groups, Geom. Topol. 16 (2012), no. 2, 781--888

    2012

  20. [28]

    Brian Bowditch, The E nding L amination T heorem , https://homepages.warwick.ac.uk/ masgak/papers/elt-fourth.pdf

  21. [29]

    Federico Berlai and Bruno Robbio, A refined combination theorem for hierarchically hyperbolic groups, Groups Geom. Dyn. 14 (2020), no. 4, 1127--1203

    2020

  22. [30]

    Daniel Berlyne and Jacob Russell, Hierarchical hyperbolicity of graph products, Groups Geom. Dyn. 16 (2022), no. 2, 523--580

    2022

  23. [31]

    Serván, Isometric embeddings of T eichm\"uller spaces are covering constructions , 2023, arXiv:2305.04153

    Frederik Benirschke and Carlos A. Serván, Isometric embeddings of T eichm\"uller spaces are covering constructions , 2023, arXiv:2305.04153

    2023 arXiv

  24. [32]

    Marissa Chesser, Stable subgroups of the G enus T wo H andlebody G roup , 2020, arXiv:2009.05067

    2020 arXiv

  25. [33]

    Dawei Chen, Dynamical invariants and intersection theory on the flex and gothic loci, Eur. J. Math. 8 (2022), no. suppl. 1, S42--S52

    2022

  26. [34]

    Dawei Chen, Martin M\" o ller, and Adrien Sauvaget, Masur- V eech volumes and intersection theory: the principal strata of quadratic differentials , Duke Math. J. 172 (2023), no. 9, 1735--1779

    2023

  27. [35]

    Aaron Calderon and Jacob Russell, Hierarchical hyperbolicity of admissible curve graphs and the boundary of marked strata, 2024, arXiv:2409.06798

    2024 arXiv

  28. [36]

    Dawei Chen and Alex Wright, The WYSIWYG compactification , J. Lond. Math. Soc. (2) 103 (2021), no. 2, 490--515

    2021

  29. [37]

    Durham, Christopher J

    Spencer Dowdall, Matthew G. Durham, Christopher J. Leininger, and Alessandro Sisto, Extensions of V eech groups II : H ierarchical hyperbolicity and quasi-isometric rigidity , Comment. Math. Helv. 99 (2024), no. 1, 149--228

    2024

  30. [38]

    Spencer Dowdall, Moon Duchin, and Howard Masur, Statistical hyperbolicity in T eichm\" u ller space , Geom. Funct. Anal. 24 (2014), no. 3, 748--795

    2014

  31. [39]

    Georgios Daskalopoulos and Chikako Mese, Infinite energy maps and rigidity, 2021, arXiv:2112.13961

    2021 arXiv

  32. [40]

    Spencer Dowdall and Howard Masur, Lattice point counting for finite-order mapping classes, 2023, arXiv:2305.03721

    2023

  33. [41]

    Durham, Yair N

    Matthew G. Durham, Yair N. Minsky, and Alessandro Sisto, Stable cubulations, bicombings, and barycenters, 2020, arXiv:2009.13647

    2020 arXiv

  34. [42]

    Taylor, The co-surface graph and the geometry of hyperbolic free group extensions, J

    Spencer Dowdall and Samuel J. Taylor, The co-surface graph and the geometry of hyperbolic free group extensions, J. Topol. 10 (2017), no. 2, 447--482

    2017

  35. [43]

    Matthew Gentry Durham, The augmented marking complex of a surface, J. Lond. Math. Soc. (2) 94 (2016), no. 3, 933--969

    2016

  36. [44]

    Alex Eskin, Simion Filip, and Alex Wright, The algebraic hull of the K ontsevich- Z orich cocycle , Ann. of Math. (2) 188 (2018), no. 1, 281--313

    2018

  37. [45]

    Alex Eskin, Maxim Kontsevich, and Anton Zorich, Sum of L yapunov exponents of the H odge bundle with respect to the T eichm\" u ller geodesic flow , Publ. Math. Inst. Hautes \' E tudes Sci. 120 (2014), 207--333

    2014

  38. [46]

    Alex Eskin and Maryam Mirzakhani, Invariant and stationary measures for the SL (2, R) action on moduli space , Publ. Math. Inst. Hautes \' E tudes Sci. 127 (2018), 95--324

    2018

  39. [47]

    Alex Eskin, Maryam Mirzakhani, and Amir Mohammadi, Isolation, equidistribution, and orbit closures for the SL (2, R) action on moduli space , Ann. of Math. (2) 182 (2015), no. 2, 673--721

    2015

  40. [48]

    McMullen, Ronen E

    Alex Eskin, Curtis T. McMullen, Ronen E. Mukamel, and Alex Wright, Billiards, quadrilaterals and moduli spaces, J. Amer. Math. Soc. 33 (2020), no. 4, 1039--1086

    2020

  41. [49]

    Alex Eskin, Howard Masur, and Kasra Rafi, Large-scale rank of T eichm\" u ller space , Duke Math. J. 166 (2017), no. 8, 1517--1572

    2017

  42. [50]

    Simion Filip, Splitting mixed H odge structures over affine invariant manifolds , Ann. of Math. (2) 183 (2016), no. 2, 681--713

    2016

  43. [51]

    David Futer and Saul Schleimer, Cusp geometry of fibered 3-manifolds, Amer. J. Math. 136 (2014), no. 2, 309--356

    2014

  44. [52]

    Dmitri Gekhtman and Mark Greenfield, Isometric submersions of T eichm\" u ller spaces are forgetful , Israel J. Math. 247 (2022), no. 2, 499--517

    2022

  45. [53]

    430, S\' e minaire Bourbaki

    \' E lise Goujard, Sous-vari\' e t\' e s totalement g\' e od\' e siques des espaces de modules de R iemann [d'apr\`es E skin, M c M ullen, M ukamel, W right] , Ast\' e risque (2021), no. 430, S\' e minaire Bourbaki. Vol. 2019/2021. Expos\' e s 1166--1180, Exp. No. 1178, 407--424

    2021

  46. [54]

    Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent

    John L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157--176

    1986

  47. [55]

    Allen Hatcher, Vector B undles and K - T heory , https://pi.math.cornell.edu/ hatcher/VBKT/VB.pdf

  48. [56]

    Thomas Haettel, Nima Hoda, and Harry Petyt, Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity, Geom. Topol. 27 (2023), no. 4, 1587--1633

    2023

  49. [57]

    Mark Hagen, Alexandre Martin, and Alessandro Sisto, Extra-large type A rtin groups are hierarchically hyperbolic , Math. Ann. 388 (2024), no. 1, 867--938

    2024

  50. [58]

    Mark Hagen, Jacob Russell, Alessandro Sisto, and Davide Spriano, Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms, 2022, arXiv:2206.12244

    2022 arXiv

  51. [59]

    Sam Hughes, Lattices in a product of trees, hierarchically hyperbolic groups and virtual torsion-freeness, Bull. Lond. Math. Soc. 54 (2022), no. 4, 1413--1419

    2022

  52. [60]

    Steven Kerckhoff, Howard Masur, and John Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), no. 2, 293--311

    1986

  53. [61]

    Michael Kopreski, Multiarc and curve graphs are hierarchically hyperbolic, 2023, arXiv:2311.04356

    2023 arXiv

  54. [62]

    Jeremy Kahn and Alex Wright, Hodge and T eichm\" u ller , J. Mod. Dyn. 18 (2022), 149--160

    2022

  55. [63]

    Stanis aw ojasiewicz, Introduction to complex analytic geometry, Birkh\" a user Verlag, Basel, 1991, Translated from the Polish by Maciej Klimek

    1991

  56. [64]

    Differential Geom

    Anna Lenzhen and Kasra Rafi, Length of a curve is quasi-convex along a T eichm\" u ller geodesic , J. Differential Geom. 88 (2011), no. 2, 267--295

    2011

  57. [65]

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

    1985

  58. [66]

    McMullen, Dynamics of SL _2( R) over moduli space in genus two , Ann

    Curtis T. McMullen, Dynamics of SL _2( R) over moduli space in genus two , Ann. of Math. (2) 165 (2007), no. 2, 397--456

    2007

  59. [67]

    , Rigidity of T eichm\" u ller curves , Math. Res. Lett. 16 (2009), no. 4, 647--649

    2009

  60. [68]

    , Billiards and T eichm\" u ller curves , Bull. Amer. Math. Soc. (N.S.) 60 (2023), no. 2, 195--250

    2023

  61. [69]

    Minsky, Harmonic maps, length, and energy in T eichm\" u ller space , J

    Yair N. Minsky, Harmonic maps, length, and energy in T eichm\" u ller space , J. Differential Geom. 35 (1992), no. 1, 151--217

    1992

  62. [70]

    , Extremal length estimates and product regions in T eichm\" u ller space , Duke Math. J. 83 (1996), no. 2, 249--286

    1996

  63. [71]

    Masur and Yair N

    Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I . H yperbolicity , Invent. Math. 138 (1999), no. 1, 103--149

    1999

  64. [72]

    H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. II . H ierarchical structure , Geom. Funct. Anal. 10 (2000), no. 4, 902--974

    2000

  65. [73]

    McMullen, Ronen E

    Curtis T. McMullen, Ronen E. Mukamel, and Alex Wright, Cubic curves and totally geodesic subvarieties of moduli space, Ann. of Math. (2) 185 (2017), no. 3, 957--990

    2017

  66. [74]

    Reine Angew

    Yair Minsky and Barak Weiss, Nondivergence of horocyclic flows on moduli space, J. Reine Angew. Math. 552 (2002), 131--177

    2002

  67. [75]

    Maryam Mirzakhani and Alex Wright, The boundary of an affine invariant submanifold, Invent. Math. 209 (2017), no. 3, 927--984

    2017

  68. [76]

    Group Theory 27 (2024), no

    Hoang Thanh Nguyen and Yulan Qing, Sublinearly M orse boundary of CAT(0) admissible groups , J. Group Theory 27 (2024), no. 4, 857--897

    2024

  69. [77]

    Duc-Manh Nguyen and Alex Wright, Non- V eech surfaces in H ^ hyp (4) are generic , Geom. Funct. Anal. 24 (2014), no. 4, 1316--1335

    2014

  70. [78]

    Harry Petyt, Mapping class groups are quasicubical, 2021, arXiv:2112.10681

    2021 arXiv

  71. [79]

    Kasra Rafi, A combinatorial model for the T eichm\" u ller metric , Geom. Funct. Anal. 17 (2007), no. 3, 936--959

    2007

  72. [80]

    , Thick-thin decomposition for quadratic differentials, Math. Res. Lett. 14 (2007), no. 2, 333--341

    2007

  73. [81]

    , Hyperbolicity in T eichm\" u ller space , Geom. Topol. 18 (2014), no. 5, 3025--3053

    2014

  74. [82]

    London Math

    Mary Rees, Teichm\" u ller distance is not C^ 2+ , Proc. London Math. Soc. (3) 88 (2004), no. 1, 114--134

    2004

  75. [83]

    H. L. Royden, Automorphisms and isometries of T eichm\" u ller space , Advances in the T heory of R iemann S urfaces ( P roc. C onf., S tony B rook, N . Y ., 1969), Ann. of Math. Stud., No. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 369--383

    1969

  76. [84]

    Kasra Rafi and Saul Schleimer, Covers and the curve complex, Geom. Topol. 13 (2009), no. 4, 2141--2162

    2009

  77. [85]

    Jacob Russell, Davide Spriano, and Hung Cong Tran, Convexity in hierarchically hyperbolic spaces, Algebr. Geom. Topol. 23 (2023), no. 3, 1167--1248

    2023

  78. [86]

    Jacob Russell, Extensions of multicurve stabilizers are hierarchically hyperbolic, 2021, arXiv:2107.14116

    2021

  79. [87]

    Alessandro Sisto, What is a hierarchically hyperbolic space?, Beyond hyperbolicity, London Math. Soc. Lecture Note Ser., vol. 454, Cambridge Univ. Press, Cambridge, 2019, pp. 117--148

    2019

  80. [88]

    5, Springer-Verlag, Berlin, 1984

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

    1984

  81. [89]

    Dedicata 161 (2012), 233--237

    Robert Tang, The curve complex and covers via hyperbolic 3-manifolds, Geom. Dedicata 161 (2012), 233--237

    2012

  82. [90]

    , Projections in the curve complex arising from covering maps, Pacific J. Math. 291 (2017), no. 1, 213--239

    2017

  83. [91]

    , Affine diffeomorphism groups are undistorted, J. Lond. Math. Soc. (2) 104 (2021), no. 2, 747--769

    2021

  84. [92]

    David Torres-Teigell, Masur- V eech volume of the gothic locus , J. Lond. Math. Soc. (2) 102 (2020), no. 1, 405--436

    2020

  85. [93]

    Robert Tang and Richard C. H. Webb, Shadows of T eichm\" u ller discs in the curve graph , Int. Math. Res. Not. IMRN (2018), no. 11, 3301--3341

    2018

  86. [94]

    Vokes, Hierarchical hyperbolicity of graphs of multicurves, Algebr

    Kate M. Vokes, Hierarchical hyperbolicity of graphs of multicurves, Algebr. Geom. Topol. 22 (2022), no. 1, 113--151

    2022

  87. [95]

    Hassler Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496--549

    1965

  88. [96]

    Karl Winsor, Saturated orbit closures in the H odge bundle , Int. Math. Res. Not. IMRN (2024), no. 6, 4710--4737

    2024

  89. [97]

    Wolpert, The hyperbolic metric and the geometry of the universal curve, J

    Scott A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), no. 2, 417--472

    1990

  90. [98]

    113, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2010

    , Families of R iemann surfaces and W eil- P etersson geometry , CBMS Regional Conference Series in Mathematics, vol. 113, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2010

    2010

  91. [99]

    Alex Wright, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015), no. 1, 413--438

    2015

  92. [100]

    , Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surv. Math. Sci. 2 (2015), no. 1, 63--108

    2015

  93. [101]

    Differential Geom

    , Totally geodesic submanifolds of T eichm\" u ller space , J. Differential Geom. 115 (2020), no. 3, 565--575

    2020

  94. [102]

    Dedicata 217 (2023), no

    Florent Ygouf, Nonarithmetic affine invariant orbifolds in H^ odd (2,2) and H(3,1) , Geom. Dedicata 217 (2023), no. 4, Paper No. 72, 24

    2023

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