Pith. sign in

REVIEW 2 major objections 49 references

Simple-stable representations of surface groups in $\mathrm{PU}(2,1)$

T0 review · 2 major / 0 minor · reviewed 2026-06-29 · grok-4.3

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

Pith's one-line read The conjugacy classes of simple-stable representations of surface groups into PU(2,1) form a domain of discontinuity strictly larger than the convex cocompact ones.

desk verdict The paper defines simple-stable representations modeled on primitive-stable ones and claims they form a strictly larger domain of discontinuity for Out(Γ_g) on X(Γ_g, PU(2,1)) than the convex cocompact representations. read the letter →

arxiv 2605.28891 v1 pith:LXVPENT6 submitted 2026-05-27 math.GT

Ulysse Remfort-Aurat (I2M) This is my paper · ORCID
classification math.GT
keywords simple-stablerepresentationssurfacegroupsPU(21)charactervarietydomainofdiscontinuityouterautomorphismgroupconvexcocompact
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 establishes that for the fundamental group of a closed orientable surface of genus at least two, the conjugacy classes of simple-stable representations into PU(2,1) constitute a domain of discontinuity for the natural action of the outer automorphism group. This set properly contains the conjugacy classes of convex cocompact representations. A sympathetic reader would care because domains of discontinuity organize the structure of character varieties and reveal where the outer automorphism action behaves properly. The definition of simple-stable representations is modeled directly on Minsky's primitive-stable representations but adapted to allow a strictly bigger class.

What carries the argument

Simple-stable representations, defined by analogy with primitive-stable representations to ensure the discontinuity property holds for the outer automorphism action.

What would settle it

A sequence of pairwise non-conjugate simple-stable representations whose conjugacy classes converge to a limit point that remains inside the claimed domain.

Watch

Extended reading notes

Core claim

We prove that the set of conjugacy classes of simple-stable representations of Γ_g in PU(2,1) is a domain of discontinuity for the Out(Γ_g) action on the character variety, strictly larger than the set of conjugacy classes of convex cocompact representations.

Load-bearing premise

The definition of simple-stable representations can be made precise enough for the discontinuity proof to hold without extra unstated conditions on the representations.

Editorial extensions

If this is right

  • The outer automorphism group acts properly discontinuously on a strictly larger open subset of the character variety than was previously known.
  • Convex cocompact representations form a proper subset of the simple-stable ones.
  • New open sets exist in the PU(2,1) character variety where the dynamics of the outer automorphism action are controlled.
  • The stability condition provides a systematic way to enlarge known discontinuity domains for surface group representations.

Reading between the lines

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

  • Analogous stability notions might produce larger discontinuity domains for representations into other groups such as PU(n,1) for n>2.
  • The boundary between simple-stable and non-simple-stable representations could be studied by examining limiting behavior of specific sequences.
  • This construction may connect to questions about the topology of the full character variety by identifying larger regions of controlled dynamics.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The paper defines simple-stable representations of the surface group Γ_g (g≥2) into PU(2,1), modeled on Minsky's primitive-stable representations. It proves that the set S of conjugacy classes of such representations forms a domain of discontinuity for the Out(Γ_g) action on the character variety X(Γ_g, PU(2,1)), and that S properly contains the set of convex cocompact representations.

Significance. If the central claim holds, the result would identify a strictly larger domain of discontinuity than the convex cocompact locus in the PU(2,1) character variety. This extends the theory of Out(Γ_g)-domains of discontinuity from real hyperbolic settings to complex hyperbolic geometry and could inform the study of proper actions and geometric invariants such as the Cartan invariant.

major comments (2)
  1. [§2 (definition)] Definition of simple-stable representations (modeled in the introduction and §2): the definition requires only that simple closed curves map to non-elliptic isometries. This is insufficient to guarantee proper discontinuity, because PU(2,1) isometries include parabolics and the action involves the Cartan invariant; without an explicit uniform lower bound on translation length (or an equivalent discreteness criterion) the set S may fail to be open or the action may fail to be properly discontinuous, as sequences can accumulate at the boundary of X while Out elements escape.
  2. [main theorem / §4] Proof that S is a domain of discontinuity (main theorem, presumably §4 or Theorem 1.1): the argument must derive openness of S and proper discontinuity directly from the definition. If the proof relies on an implicit discreteness property not stated in the definition, the load-bearing step is missing; the abstract gives no indication whether discreteness is proved or assumed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points that require clarification. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§2 (definition)] Definition of simple-stable representations (modeled in the introduction and §2): the definition requires only that simple closed curves map to non-elliptic isometries. This is insufficient to guarantee proper discontinuity, because PU(2,1) isometries include parabolics and the action involves the Cartan invariant; without an explicit uniform lower bound on translation length (or an equivalent discreteness criterion) the set S may fail to be open or the action may fail to be properly discontinuous, as sequences can accumulate at the boundary of X while Out elements escape.

    Authors: The referee correctly notes that the definition as written in §2 only requires images of simple closed curves to be non-elliptic. This formulation is modeled directly on the non-elliptic condition in Minsky's primitive-stable representations but does not explicitly encode a uniform lower bound on translation length. We agree that, without such a bound, openness of S and proper discontinuity of the Out(Γ_g) action are not immediate, particularly given the possible presence of parabolics and the role of the Cartan invariant. We will revise the definition in §2 to include an explicit uniform lower bound on translation lengths (or an equivalent discreteness criterion) for the images of all simple closed curves. This change will be used throughout the subsequent arguments. revision: yes

  2. Referee: [main theorem / §4] Proof that S is a domain of discontinuity (main theorem, presumably §4 or Theorem 1.1): the argument must derive openness of S and proper discontinuity directly from the definition. If the proof relies on an implicit discreteness property not stated in the definition, the load-bearing step is missing; the abstract gives no indication whether discreteness is proved or assumed.

    Authors: The proof in §4 establishes openness of S and proper discontinuity of the Out(Γ_g) action by using the (revised) definition to obtain a uniform lower bound on translation lengths, which in turn yields a discreteness criterion for the representations. The Cartan invariant is controlled via the non-elliptic condition together with the length bound. All steps are explicit in the body of the paper; no implicit assumption is used. The abstract is a high-level summary and does not enumerate proof details, but the full argument appears in §4. With the definitional revision noted above, the load-bearing steps will be stated directly from the definition. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected in derivation

full rationale

The paper defines simple-stable representations by direct analogy to Minsky's primitive-stable representations and proves that their conjugacy classes form a domain of discontinuity for the Out(Γ_g) action on X(Γ_g, PU(2,1)), strictly larger than the convex cocompact locus. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. The central claim is presented as an independent theorem whose proof is not shown to collapse into its inputs by construction. The modeling citation to Minsky is external and does not bear the load of the discontinuity statement itself.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Only the abstract is available, so the ledger reflects the minimal information; no free parameters or invented entities are apparent from the abstract.

assumptions (1)
  • standard math Standard properties of fundamental groups of surfaces and Lie groups like PU(2,1).
    The paper relies on background from geometric group theory and complex hyperbolic geometry.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Simple-stable representations of surface groups in $\mathrm{PU}(2,1)$." pith.science (2026). https://pith.science/paper/LXVPENT6

@misc{pith2026260528891,
  author       = {Pith},
  title        = {Pith review of: Simple-stable representations of surface groups in $\mathrmPU(2,1)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXVPENT6}},
  note         = {Machine review of arXiv:2605.28891}
}
abstract

Let $\Gamma_g$ be the fundamental group of a closed orientable surface of genus $g\geqslant 2$. The outer automorphism group $\mathrm{Out}(\Gamma_g)$ naturally acts on the character variety $\mathcal{X}(\Gamma_g,G)$ for any Lie group $G$. We consider the set of simple-stable representations which are modelled on Minsky's primitive-stable representations. We prove that the set of conjugacy classes of simple-stable representations of $\Gamma_g$ in $\mathrm{PU}(2,1)$ is a domain of discontinuity for this action, strictly larger than the set of conjugacy classes of convex cocompact representations.

Figures

Figures reproduced from arXiv: 2605.28891 by the authors.

Figure 1
Figure 1. An invariant family of horoballs of H2 . Lemma 2.7 Let C ⩾ 0 and (Rn)n∈N be a sequence of real numbers such that Rn → +∞. If (Qn)n∈N = ({Hn p }p∈P)n∈N is a sequence of Rn-separated, C-quasi-invariant families of horoballs, then for all p ∈ P, the radius of Hn p tends to 0 as n tends to ∞. Proof. Fix a basepoint o ∈ X, let p ∈ P be a parabolic fixed point and let xn be a closest point projection of o on Hn p , i.e., … view at source ↗
Figure 2
Figure 2. The geodesics Lh, g0(Lh), Ln and h k0 g0h −k0 (Ln). 17 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. The sequences (qn)n∈N and ρ(γn) +  n∈N . Claim: Up to replacing γn by one of its conjugates, we can suppose that there is a compact subset K of Λ [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: The different values of tr(A) ∈ C with the deltoid ∆. In contrast to PO0(3, 1) ≃ PSL2(C) ≃ Isom+(H3 ), the subset of parabolic elements in PU(2, 1) has real codimension 1 (instead of 2) and there exist parabolic isometries which are not unipotent. Moreover, the subset …
Figure 5
Figure 5. Figure 5: The unit Cygan sphere defined by (x 2 + y 2 ) 2 + z 2 = 1 and the complex plane Two bisectors B1 and B2 are said to be coequidistant if there are z, w1, w2 ∈ H2 C such that B1 = B(z, w1) and B2 = (z, w2). Proposition 5.5 If B1 and B2 are intersecting, infinite, coequid…
Figure 6
Figure 6. Figure 6: A fundamental domain D and its image under the elements of Gz (z = ∞) The set Gz must be infinite. Indeed, if Gz is finite, there is a neighbourhood of z in H2 C which meets only finitely many Γ-translates of D, contradicting the fact that z ∈ Λ(Γ). Using Claim 1, we d…
Figure 7
Figure 7. Figure 7: A finite half-space bounded by a finite bisector [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]
Figure 8
Figure 8. Figure 8: A geodesic ray β, the side S and the sequence (yn)n∈N First, suppose that a ̸= y and γ −1 n (z) → y. Since yn → y and a ̸= y, the sequence (yn)n∈N is contained in a compact subset of H2 C \ {a}, therefore γn(yn) → b. Moreover, for all n ∈ N, γn(yn) ∈ β = [o, z) . We de…
Figure 9
Figure 9. Figure 9: A geometric realization of ∆3,3,9 As before, we allow one or more of p, q, r to be infinite. In such cases, we replace the concurrent complex lines by asymptotic ones. In contrast to real hyperbolic geometry, when p ⩾ 3, there is a (real) one-parameter family of non-is…
Figure 10
Figure 10. Figure 10: The representative curves of x 7→ 2x 2+1 3x in green and x 7→ 12x 2−1 12x in red. We give a more accurate description of the situation: Proposition 6.8 Let n be an integer at least 4. • For every α ∈ (α 0 n , π], a (3, 3, n; α) representation of ∆3,3,n in PU(2, 1) is …
Figure 11
Figure 11. Figure 11: The quadrilateral ABCD. Lemma 6.11 The following statements are equivalent: 1) d(A, D) = d(B, C) 2) dH1 (A, D) = dH2 (B, C) 3) ABCD is invariant under the inversion If in Lf . where dH1 and dH2 denote the arc length distances along H1 and H2 respectively. Moreover, if…
Figure 12
Figure 12. Figure 12: The 18-gon P with relevant side pairings There are exactly six vertex cycles, each of which have angle sum equal to 2π. The group Γ generated by S = {s ± I , ..., s± IX} is isomorphic to the fundamental group of a closed oriented surface of genus 2 by Theorem 6.9 and …
Figure 13
Figure 13. Figure 13: The geodesic Lf and its image in the genus 2 surface Σ = P/ ∼S 39 [PITH_FULL_IMAGE:figures/full_fig_p039_13.png]
Figure 14
Figure 14. Figure 14: The cyclic cover p β g when g = 5. We can also describe p β g algebraically. Denote by ˆı(β, γ) the algebraic intersection number be￾tween the closed curves β and γ. The subgroup of π1(Σ2) associated to p β g consists of free homotopy classes of closed curves on Σ2 wh…
Figure 15
Figure 15. Figure 15: An example of a curve β. References [1] Boris N. Apanasov. Dynamics of discrete group action, volume 10 of Adv. Anal. Geom. Berlin: De Gruyter, 2024. [2] Alan F. Beardon. The geometry of discrete groups, volume 91 of Grad. Texts Math. Springer, Cham, 1983. [3] Alan F.…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 4 canonical work pages

  1. [1]

    Apanasov

    Boris N. Apanasov. Dynamics of discrete group action, volume 10 ofAdv. Anal. Geom. Berlin: De Gruyter, 2024

    2024

  2. [2]

    Beardon.The geometry of discrete groups, volume 91 ofGrad

    Alan F. Beardon.The geometry of discrete groups, volume 91 ofGrad. Texts Math.Springer, Cham, 1983

    1983

  3. [3]

    Beardon and Bernard Maskit

    Alan F. Beardon and Bernard Maskit. Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta Math., 132:1–12, 1974

    1974

  4. [4]

    Spaces of Kleinian groups

    Lipman Bers. Spaces of Kleinian groups. Several Complex Variables I, Conf. Univ. Maryland 1970, Lect. Notes Math. 155, 9-34 (1970)., 1970

    1970

  5. [5]

    B. H. Bowditch. Discrete parabolic groups.J. Differ. Geom., 38(3):559–583, 1993

    1993

  6. [6]

    B. H. Bowditch. Geometrical finiteness with variable negative curvature.Duke Math. J., 77(1):229–274, 1995

    1995

  7. [7]

    B. H. Bowditch. Markoff triples and quasifuchsian groups. Proc. Lond. Math. Soc. (3), 77(3):697–736, 1998

    1998

  8. [8]

    B. H. Bowditch. Relatively hyperbolic groups.Int. J. Algebra Comput., 22(3):1250016, 66, 2012

    2012

Show all 49 references
  1. [9]

    A global shadow lemma and logarithm law for geometri- cally finite Hilbert geometries

    Harrison Bray and Giulio Tiozzo. A global shadow lemma and logarithm law for geometri- cally finite Hilbert geometries. Preprint, arXiv:2111.04618 [math.DS] (2021), 2021. 42

    2021

  2. [10]

    Bridson and André Haefliger.Metric spaces of non-positive curvature, volume 319 ofGrundlehren Math

    Martin R. Bridson and André Haefliger.Metric spaces of non-positive curvature, volume 319 ofGrundlehren Math. Wiss.Berlin: Springer, 1999

    1999

  3. [11]

    Almost-fuchsian representations in PU(2,1)

    Samuel Bronstein. Almost-fuchsian representations in PU(2,1). Preprint, arXiv:2411.16261 [math.DG] (2024), 2024

    2024

  4. [12]

    D. R. J. Chillingworth. Simple closed curves on surfaces.Bull. Lond. Math. Soc., 1:310–314, 1969

    1969

  5. [13]

    Les groupes hyperboliques de Gromov

    Michel Coornaert, Thomas Delzant, and Athanase Papadopoulos.Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov. (Geometry and group theory. The hyperbolic groups of Gromov), volume 1441 ofLect. Notes Math.Berlin etc.: Springer-Verlag, 1990

    1990

  6. [14]

    Benson Farb and Dan Margalit.A primer on mapping class groups, volume 49 ofPrinceton Math. Ser. Princeton, NJ: Princeton University Press, 2011

    2011

  7. [15]

    William J. Floyd. Group completions and limit sets of Kleinian groups. Invent. Math., 57:205–218, 1980

    1980

  8. [16]

    F. W. Gehring and G. J. Martin. Discrete quasiconformal groups. I.Proc. Lond. Math. Soc. (3), 55:331–358, 1987

    1987

  9. [17]

    Floyd maps for relatively hyperbolic groups

    Victor Gerasimov. Floyd maps for relatively hyperbolic groups. Geom. Funct. Anal., 22(5):1361–1399, 2012

    2012

  10. [18]

    Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups.J

    Victor Gerasimov and Leonid Potyagailo. Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups.J. Eur. Math. Soc. (JEMS), 15(6):2115–2137, 2013

    2013

  11. [19]

    Hyperbolic groups

    Étienne Ghys. Hyperbolic groups. Sémin. Bourbaki, Vol. 1989/90, 42ème année, Astérisque 189-190, Exp. No. 722, 203-238 (1990)., 1990

    1989

  12. [20]

    (On the hyperbolic groups à la M

    Etienne Ghys and Pierre de la Harpe, editors.Sur les groupes hyperboliques d’après Mikhael Gromov. (On the hyperbolic groups à la M. Gromov), volume 83 ofProg. Math. Boston, MA: Birkhäuser, 1990

    1990

  13. [21]

    William M. Goldman. Topological components of spaces of representations.Invent. Math., 93(3):557–607, 1988

    1988

  14. [22]

    Goldman.Complex hyperbolic geometry

    William M. Goldman.Complex hyperbolic geometry. Oxford Math. Monogr. Oxford: Claren- don Press, 1999

    1999

  15. [23]

    William M. Goldman. Mapping class group dynamics on surface group representations. In Problems on mapping class groups and related topics, pages 189–214. Providence, RI: American Mathematical Society (AMS), 2006

    2006

  16. [24]

    Complexhyperbolicmanifolds homotopy equivalent to a Riemann surface.Commun

    WilliamM.Goldman, MichaelKapovich, andBernhardLeeb. Complexhyperbolicmanifolds homotopy equivalent to a Riemann surface.Commun. Anal. Geom., 9(1):61–95, 2001

    2001

  17. [25]

    William. M. Goldman and John. R. Parker. Complex hyperbolic ideal triangle groups.J. Reine Angew. Math., 425:71–86, 1992

    1992

  18. [26]

    A survey of complex hyperbolic Kleinian groups

    Michael Kapovich. A survey of complex hyperbolic Kleinian groups. InIn the tradition of Thurston II. Geometry and groups, pages 7–51. Cham: Springer, 2022

    2022

  19. [27]

    Ashortanddirtyintroductiontohyperbolicsurfaces

    FrançoisLabourie. Ashortanddirtyintroductiontohyperbolicsurfaces. https://math.univ- cotedazur.fr/ labourie/preprints/pdf/HypGeom.pdf, 2012. 43

    2012

  20. [28]

    Dynamics on PSL(2, C)-character varieties of certain hyperbolic 3-manifolds

    Michelle Dongeun Lee. Dynamics on PSL(2, C)-character varieties of certain hyperbolic 3-manifolds. PhD thesis, University of Michigan, 2012

    2012

  21. [29]

    The modular action onPSL2(R)-characters in genus 2

    Julien Marché and Maxime Wolff. The modular action onPSL2(R)-characters in genus 2. Duke Math. J., 165(2):371–412, 2016

    2016

  22. [30]

    Yair N. Minsky. On dynamics ofOut(Fn) on PSL2(C) characters. Isr. J. Math., 193:47–70, 2013

    2013

  23. [31]

    John R. Parker. Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space. Geom. Dedicata, 57(3):223–234, 1995

    1995

  24. [32]

    John R. Parker. Notes on complex hyperbolic geometry. https://maths.dur.ac.uk/users/j.r.parker/img/NCHG.pdf, 2003

    2003

  25. [33]

    Parker and Ioannis D

    John R. Parker and Ioannis D. Platis. Complex hyperbolic quasi-Fuchsian groups. In Geometry of Riemann surfaces. Proceedings of the Anogia conference to celebrate the 65th birthday of William J. Harvey, Anogia, Crete, Greece, June–July 2007, pages 309–355. Cambridge: Cambridge...

    2007

  26. [34]

    Parker, Jieyan Wang, and Baohua Xie

    John R. Parker, Jieyan Wang, and Baohua Xie. Complex hyperbolic(3, 3, n)triangle groups. Pac. J. Math., 280(2):433–453, 2016

    2016

  27. [35]

    Traces in complex hyperbolic triangle groups

    Anna Pratoussevitch. Traces in complex hyperbolic triangle groups. Geom. Dedicata, 111:159–185, 2005

    2005

  28. [36]

    On some stable representations of hyperbolic groups

    Ulysse Remfort-Aurat. On some stable representations of hyperbolic groups. Preprint, arXiv:2310.19329 [math.GT] (2023), 2023

    2023

  29. [37]

    Ideal triangle groups, dented tori, and numerical analysis.Ann

    Richard Evan Schwartz. Ideal triangle groups, dented tori, and numerical analysis.Ann. Math. (2), 153(3):533–598, 2001

    2001

  30. [38]

    Complex hyperbolic triangle groups

    Richard Evan Schwartz. Complex hyperbolic triangle groups. In Proceedings of the in- ternational congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures, pages 339–349. Beijing: Higher Education Press; Singapore: World Scientific/dis...

    2002

  31. [39]

    A better proof of the Goldman-Parker conjecture.Geom

    Richard Evan Schwartz. A better proof of the Goldman-Parker conjecture.Geom. Topol., 9:1539–1601, 2005

    2005

  32. [40]

    Spherical CR geometry and Dehn surgery, volume 165 of Ann

    Richard Evan Schwartz. Spherical CR geometry and Dehn surgery, volume 165 of Ann. Math. Stud. Princeton, NJ: Princeton University Press, 2007

    2007

  33. [41]

    Hyperbolic geometry

    Caroline Series. Hyperbolic geometry. https://warwick.ac.uk/fac/sci/maths/people/staff- /caroline_series/hyperbolic_geometry_ma448_lecture_notes.pdf, 2008

    2008

  34. [42]

    Smogorzhevsky

    A.S. Smogorzhevsky. Lobachevskian geometry. Little Mathematics Library. Mir Publishers, 1982

    1982

  35. [43]

    Simple Anosov representations of closed surface groups

    Nicolas Tholozan and Tianqi Wang. Simple Anosov representations of closed surface groups. Preprint, arXiv:2307.02934 [math.GT] (2023), 2023

    2023

  36. [44]

    Representations of surface groups in complex hyperbolic space.J

    Domingo Toledo. Representations of surface groups in complex hyperbolic space.J. Differ. Geom., 29(1):125–133, 1989

    1989

  37. [45]

    Convergence groups and Gromov’s metric hyperbolic spaces.N

    Pekka Tukia. Convergence groups and Gromov’s metric hyperbolic spaces.N. Z. J. Math., 23(2):157–187, 1994. 44

    1994

  38. [46]

    Conical limit points and uniform convergence groups.J

    Pekka Tukia. Conical limit points and uniform convergence groups.J. Reine Angew. Math., 501:71–98, 1998

    1998

  39. [47]

    Gromov hyperbolic spaces.Expo

    Jussi Väisälä. Gromov hyperbolic spaces.Expo. Math., 23(3):187–231, 2005

    2005

  40. [48]

    Anosov representations over closed subflows

    Tianqi Wang. Anosov representations over closed subflows. Trans. Am. Math. Soc., 376(9):6177–6214, 2023

    2023

  41. [49]

    Eugene Z. Xia. The moduli of flat PU(2,1) structures on Riemann surfaces.Pac. J. Math., 195(1):231–256, 2000. 45

    2000

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.