Pith. sign in

REVIEW 5 minor 66 references

Bending, entropy and proper affine actions of surface groups

T0 review · 0 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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

Pith's one-line read An explicit neighborhood of the Fuchsian locus in quasifuchsian space consists of non-Fuchsian holonomies that admit proper affine actions with linear part Ad(ρ); in a larger neighborhood, all entropy critical points are Fuchsian.

desk verdict Explicit neighborhoods of the Fuchsian locus with controlled entropy and proper affine actions; the main theorems look right, with only minor numerical glitches to clean up. read the letter →

arxiv 2602.20146 v2 pith:OMZRCFUZ submitted 2026-02-23 math.GT math.DSmath.GR

Martin Bridgeman , Richard Canary , Andres Sambarino This is my paper · ORCID
classification math.GTmath.DSmath.GR MSC 57M5037D4030F4022E40
keywords quasifuchsianspacebendinglaminationproperaffineactionentropyfunctionMargulisinvariantsurfacegroupgeodesiccurrents
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, for any closed oriented surface of genus at least two, there is an explicit open neighborhood of the Fuchsian locus in quasifuchsian space where every non-Fuchsian group is the linear part of a proper affine action on the six-dimensional space sl(2,C). The geometric engine is 'moderate bending' of the two boundary components of the convex core: when both complementary Jordan domains are moderately bent, the combined infinitesimal bending direction w+ + w− uniformly contracts every geodesic length, and the associated cocycle has a Margulis invariant spectrum bounded away from zero, which forces the affine action to be proper. The same moderate-bending condition, on just one boundary component, shows that the point is not a critical point of the entropy function; an explicit larger neighborhood therefore has no non-Fuchsian entropy critical points. The neighborhoods are made concrete by roundness bounds: if the L-roundness of a bending lamination is below r(L) (inverse of x sec x on (0,1]), the domain is moderately bent, with r(1) ≈ 0.739.

What carries the argument

The key object is the moderately bent Jordan domain: a complementary domain Ω of the limit set of a quasifuchsian group such that for every bending pair (x,y) there is a circle transverse to the boundary C with L∩C={x,y}. This condition makes the imaginary part of the complex distance between axes and bending leaves have a fixed sign in the variation formula (Theorem 3.2), giving dℓγ(wν)≤0. The two-sided version uses the binding-pair property of the bending laminations (every geodesic current meets at least one of them) to upgrade the two one-sided bounds into the uniform contraction dℓγ(w)≤−Kℓγ(ρ). The explicit roundness function r(L) — inverse of y=x sec x on (0,1] and L sech L for L>1 — t

What would settle it

Take a non-Fuchsian quasifuchsian group with both bending laminations satisfying ∥β±∥_1 < 0.739 (so the paper's roundness criterion applies) and numerically compute the normalized Margulis spectrum of the cocycle associated to the combined bending vector field w+ + w−; if 0 belongs to that spectrum, Theorem 1.2 fails. A cheaper check: compute dℓγ(w+ + w−)/ℓγ(ρ) for a long closed geodesic γ in such a group; if it is sometimes positive, Theorem 1.4's uniform contraction fails.

Watch

Extended reading notes

Core claim

The central result is that proper affine surface-group actions are not sporadic: they occur for every non-Fuchsian quasifuchsian holonomy in an explicit open set of QF(S). The sufficient condition is that both boundary components of the convex core are moderately bent — for each bending pair of the boundary of the Jordan domain, a transverse circle meets the boundary exactly there. Under this condition the infinitesimal bending deformations along the two bending laminations combine into a direction w = w+ + w− along which all geodesic lengths decrease uniformly, dℓγ(w) ≤ −Kℓγ(ρ). Theorem 1.4 turns this into a cocycle whose normalized Margulis spectrum is bounded away from 0, and the properne

Load-bearing premise

The load-bearing external premise is that the two bending laminations of any non-Fuchsian quasifuchsian group bind the surface — every geodesic current has positive intersection with at least one of them — a result cited from the literature and not proved here; if binding failed, the uniform length-contraction constant K would not follow and the properness argument would collapse.

Editorial extensions

If this is right

  • In the explicit neighborhood V(S) of the Fuchsian locus, every non-Fuchsian holonomy representation admits a proper affine action on sl(2,C) with linear part Ad(ρ).
  • In the larger neighborhood U(S), every non-Fuchsian representation has nonzero entropy derivative in some direction, so the only critical points of h in U are Fuchsian.
  • The combined bending direction w=w++w− is a global length-decreasing vector field: for every closed geodesic γ, dℓγ(w) ≤ −Kℓγ(ρ), a geometric rigidity that is the engine of the properness result.
  • Through the principal embedding, the quasifuchsian examples yield an open set of proper affine actions on the Lie algebra of any complex simple Lie group, and an open set of pairs of representations acting properly on the group manifold by left/right multiplication.
  • Quantitative criteria from Section 9: entropy-criticality and proper affine actions follow from Schwarzian norm < 0.0739, Teichmüller distance < 0.049, or quasicircle constant K < 1.05.

Reading between the lines

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

  • Inference: The uniform contraction inequality is stronger than what is needed for properness alone; it suggests that the combined bending flow may push the entire nearby deformation space away from any entropy local maxima, so the set where the entropy gradient vanishes could be exactly the Fuchsian locus well beyond the explicit neighborhood U.
  • Inference: The threshold r(L) being the inverse of x sec x (with r(1)≈0.739, the fixed point of cos) invites a numerical experiment: compute, for the paper's own horocycle-based pleated planes, the actual maximal roundness before embedding is lost; if that value exceeds r(L) while moderate bending persists, the sufficient bound is not sharp and the neighborhoods could be enlarged.
  • Inference: Because the proof passes through the binding-pair theorem for bending laminations, any surface-group deformation theory that preserves the binding property (e.g., small perturbations inside the character variety) should inherit the same properness conclusion; this predicts a whole open cone of cocycles, not just the single bending vector field, with 0 outside the Margulis spectrum.
  • Inference: The principal-embedding step is a general transfer principle: any complex simple Lie group whose principal sl(2,C)-triple has the same weight scaling inherits proper affine actions from quasifuchsian surface groups, so the phenomenon is not special to dimension 6.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper studies quasifuchsian representations of a closed surface group. It introduces a notion of 'moderately bent' Jordan domain and proves that if one domain of discontinuity is moderately bent then the representation is not a critical point of the topological entropy function (Theorem 1.1), and if both are moderately bent then the adjoint representation is the linear part of a proper affine action on the Lie algebra sl(2,C) (Theorem 1.2). The proof chain combines a formula for the variation of complex length under bending deformations (Theorem 3.2), a uniform contraction estimate for the sum of the two bending directions (Theorem 1.4), and a Margulis-spectrum properness criterion (Proposition 6.6). Using roundness bounds from Bridgeman–Canary–Yarmola, the authors produce explicit neighborhoods U(S) and V(S) of the Fuchsian locus (Corollary 1.6) and extend the results to proper affine actions on other complex simple Lie groups.

Significance. If correct, the results are significant: they provide explicit neighborhoods of the Fuchsian locus on which every non-Fuchsian representation gives a proper affine action with adjoint linear part, and on which every critical point of the entropy function is Fuchsian. This goes beyond the earlier existence results of Danciger–Guéritaud–Kassel by placing the phenomenon in an open set and giving quantitative roundness thresholds. The proof is detailed and largely self-contained; the main external inputs (Bonahon–Otal binding, Sambarino's normalized variation criterion, Kassel–Smilga properness) are standard or are proved in the text. The constants are explicit and the claims are falsifiable. I find no load-bearing gap in the central argument.

minor comments (5)
  1. [Section 1] The comparison bound displayed as '∥β±∥L ≤ 2 cos^{-1}(−sinh(L/2))' is undefined for L > 2 arcsinh(1), e.g. for L=2 the argument of cos^{-1} is less than -1. This appears only as a motivational comparison and does not affect the main results, but it should be corrected or restricted to the valid range of L.
  2. [Section 6.4, proof of Theorem 1.2] The displayed equality 'ℜ(m(ρ(g),u(g))) = (dℓγ(u), -dℓγ(u))' is missing a factor of 1/2: since ℓγ = 2 log|λ1|, one has ℜ(m) = (dℓγ/2, -dℓγ/2). The conclusion that the normalized first coordinate is bounded away from 0 is unchanged, but the formula should be corrected.
  3. [Section 9, proof of Theorem 9.1] The sentence concluding 'so our theorem holds with ϵ=.739' should read '.0739'; the preceding computation gives G(.611)≈.0739643, and the theorem statement uses .0739.
  4. [Section 5, Lemma 5.2] The inequality in the statement and proof is written with iδ(γ,β+) even when the bending lamination is βν with ν=-; it should be iδ(γ,βν) throughout.
  5. [Throughout] There are several typos: 'explict' in the abstract, 'PSL)2,C)' in the proof of Theorem 1.2, 'critcal' in the introduction, 'defornation' in the Section 3 heading, and 'U S)' in Section 8.4. Also, in Corollary 1.6 the notation 'for some L1>0 or for some L2>0' is confusing; it would be clearer to write 'for some L>0' independently for each condition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained given standard external inputs.

full rationale

I walked the paper's claimed derivation chain and found no step in which a 'prediction' or 'first-principles result' reduces by construction to its inputs. Theorem 1.3 computes dℓγ(w) directly from the Kourouniotis-style complex-length variation formula (Theorem 3.2) and the definition of moderately bent; the sign of ℑcoshσ is established from the Jordan-domain geometry, not assumed. Theorem 1.4 obtains the uniform contraction dℓγ(w) ≤ −Kℓγ(ρ) from Lemma 5.1 plus the binding property of the two bending laminations, cited from Bonahon–Otal [10, Prop. 4]; this is an external, standard theorem and is not equivalent to the conclusion. Theorem 1.5's roundness criterion is proved from the θ-bounded criterion (Theorem 8.4) and the Bridgeman–Canary–Yarmola embedding bound [15]; those ingredients do not assume the target results about entropy critical points or proper affine actions. Theorem 1.2 passes from the uniform length contraction to properness via the Margulis-invariant computation (Proposition 6.7, proved in the paper) and the Kassel–Smilga properness criterion (Proposition 6.6, also proved in the paper). The self-citations that occur (Sambarino [59], BCY [15], Bridgeman–Tee [14], etc.) are used as genuine ingredients with specific, non-target content, not as an unexamined uniqueness claim or an ansatz smuggled in by citation. In particular, Proposition 4.1 from [59] is a general criterion relating length variations to entropy variations; it does not contain the paper's conclusion that quasifuchsian points are non-critical. I therefore find no circularity.

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

The central claims require standard but heavy machinery from quasifuchsian geometry, measured laminations, Anosov representations, and thermodynamic formalism. No numerical constants are fitted to the claims; r(L), .0739, and .049 are explicit functions or chosen subthreshold values derived from cited geometric lemmas. The 'moderately bent' condition is a new technical definition rather than an independent physical entity.

assumptions (5)
  • domain assumption Bending laminations of any non-fuchsian quasifuchsian representation form a binding pair (Bonahon–Otal [10, Prop. 4]).
    Used in Theorem 1.4 to combine the two one-sided contractions via Lemma 5.1; if it failed, the uniform K in Theorem 1.4 would not follow.
  • domain assumption Sambarino's normalized-variation criterion ([59, Lemma 2.32]): for non-fuchsian ρ, -dh(v)/h(ρ) lies in the interior of {dℓγ(v)/ℓγ}, giving Proposition 4.1.
    This is the bridge from length contraction to non-criticality of entropy; the paper cites it without proof.
  • domain assumption The convex core boundary of a quasifuchsian ρ has a bending lamination, the intrinsic metric is hyperbolic, and bending deformations vary holomorphically (Kourouniotis [46,47], Theorem 3.2).
    Foundation for the vector fields wν and the complex-length variation formula on which all later estimates rest.
  • domain assumption Bridgeman–Canary–Yarmola [15] provides an explicit bilipschitz bound G(L), and Bridgeman–Tee [14] provides the bound relating Schwarzian derivative to roundness; these make the neighborhoods explicit.
    Used in Theorem 1.5 and Section 9 to turn analytic bounds into moderate bending; these are external results, one from a preprint.
  • domain assumption For a complex simple Lie group G, the principal embedding τ:PSL(2,C)→Inn(g) satisfies ϖ1(μ(τ(g))) = c_g ω1(μ(g)) (Kostant), and quasifuchsian representations lift to SL(2,C) (Culler).
    Needed for Corollary 1.7 extending proper actions to arbitrary complex simple Lie groups.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bending, entropy and proper affine actions of surface groups." pith.science (2026). https://pith.science/paper/OMZRCFUZ

@misc{pith2026260220146,
  author       = {Pith},
  title        = {Pith review of: Bending, entropy and proper affine actions of surface groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OMZRCFUZ}},
  note         = {Machine review of arXiv:2602.20146}
}
abstract

We show that for any closed surface $S$ there is an explict neighborhood $V$ of the fuchsian locus in quasifuchsian space $\mathsf{QF}(S)$ such that for every representation $\rho\in V$ which is not fuchsian, there is a proper affine action on $\mathfrak{sl}(2,\mathbb{C})$ with linear part $\mathsf{Ad}(\rho)$. We further show that there is a larger neighborhood $U$ of the Fuchsian locus so that every critical point of the entropy function in $U$ lies on the Fuchsian locus.

Figures

Figures reproduced from arXiv: 2602.20146 by the authors.

Figure 1
Figure 1. Domain Ων [u−, v−, v+, u+] = [z−, 0,∞, z+] = z+ z− = z+z− |z−| 2 . Moreover, ℑ  z+z−  = −z+ℑ(z−) > 0, so ℑ  [u−, v−, v+, u+]  > 0, which completes the proof. □ [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Plane L 8.2. Bending bounds. We now give explicit bounds on ∥µ∥L such that the associated pleated plane is θ-bounded and therefore the associated Jordan domain is moderately bounded. In earlier work [15] the first two authors and Yarmola produced an explicit upper bound on ∥µ∥L which guarantees that fµ is a bilipschitz embedding. Theorem 8.3. (Bridgeman-Canary-Yarmola [15]) There exists an explicit function G : R+ →… view at source ↗
Figure 3
Figure 3. Functions aL(θ) and rL(θ) for L ≤ 1 We obtain the following strengthening of Theorem 8.3. Theorem 8.4. (θ-bounded criterion) Let θ ∈ [0, π/2] and L > 0. If µ ∈ ML(H2 ) and ∥µ∥L < rL(θ), then Pµ is a convex pleated plane and µ is θ-bounded. In particular, if ∥µ∥L < r(L), then Pµ is θ-bounded for some θ < π 2 . Theorem 1.5 follows immediately from Proposition 8.2 and Theorem 8.4. We will discuss the proof of Corollary… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Plot of r1(θ) on [0, π 2 ] x = Lsin(θ − x). As r(L) = rL(π/2), it follows that r is the inverse function of the function L(x) = x sec(x). In particular for L = 1, r(1) = r1(π/2) ≃ 0.739085. 8.4. Proof of Corollary 1.6. Theorems 1.1 and 1.5 together imply that if ρ ∈ U(…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 5 linked inside Pith

  1. [1]

    Properly discontinuous groups of affine transformations with orthogonal linear part,

    H. Abels, G. Margulis and A. Soifer, “Properly discontinuous groups of affine transformations with orthogonal linear part,” C. R. Acad. Sci. Paris Math.324(1997), 253–258. 5, 17

    1997

  2. [2]

    The structure of complete locally affine manifolds,

    L. Auslander, “The structure of complete locally affine manifolds,”Topology3(1964), 131–139. 5

    1964

  3. [3]

    Actions propres sur les espaces homog` enes r´ eductifs,

    Y. Benoist, “Actions propres sur les espaces homog` enes r´ eductifs,”Ann. of Math.144(1996), 315–347. 5, 20

    1996

  4. [4]

    Propri´ et´ es asymptotiques des groupes lin´ eaires,

    Y. Benoist, “Propri´ et´ es asymptotiques des groupes lin´ eaires,”G.A.F.A.7(1997), 1–47. 22

    1997

  5. [5]

    Benoist and J.F

    Y. Benoist and J.F. Quint,Random Walks on Reductive Groups, Springer-Verlag, 2016. 19, 22

    2016

  6. [6]

    Simultaneous uniformization,

    L. Bers, “Simultaneous uniformization,”Bull. A.M.S.66(1960), 94–97. 30

    1960

  7. [7]

    Blayac,Aspects dynamiques des structures projectives convexes,PhD thesis, Universit` e Paris-Saclay, 2021, seehttps: //theses.hal.science/tel-0348402621

    P.-L. Blayac,Aspects dynamiques des structures projectives convexes,PhD thesis, Universit` e Paris-Saclay, 2021, seehttps: //theses.hal.science/tel-0348402621

    2021

  8. [8]

    Anosov Representations and dominated splittings,

    J. Bochi, R. Potrie, and A. Sambarino, “Anosov Representations and dominated splittings,”J. Eur. Math. Soc.21(2019), 3343–3414. 14, 16

    2019

Show all 66 references
  1. [9]

    The geometry of Teichm¨ uller space via geodesic currents,

    F. Bonahon, “The geometry of Teichm¨ uller space via geodesic currents,”Invent. Math92(1988), 139–62. 6, 12

    1988

  2. [10]

    Laminations measur´ ees de plissage des vari´ et´ es hyperboliques de dimension 3

    F. Bonahon, J.P. Otal, “Laminations measur´ ees de plissage des vari´ et´ es hyperboliques de dimension 3”,Ann. Math.,160 (2004) 1013–1055. 13

    2004

  3. [11]

    Hausdorff dimension of quasicircles

    R. Bowen, “Hausdorff dimension of quasicircles”,Publ. Math. I.H.E.S,50, (1979) 11–25. 1, 2

    1979

  4. [12]

    Hausdorff dimension and the Weil-Petersson extension to quasi-fuchsian space,

    M. Bridgeman, “Hausdorff dimension and the Weil-Petersson extension to quasi-fuchsian space,”Geom. and Top.14(2010), 799–831. 2

    2010

  5. [13]

    The pressure metric for Anosov representations,

    M. Bridgeman, D. Canary, F. Labourie, and A. Sambarino, “The pressure metric for Anosov representations,”G.A.F.A. 25(4), 2015, 1089–1179. 21

    2015

  6. [14]

    Bounds on bending in terms of the Schwartzian derivative and Teichm¨ uller distance,

    M. Bridgeman and M.H. Tee, “Bounds on bending in terms of the Schwartzian derivative and Teichm¨ uller distance,” preprint, arXiv:2509.12493. 30, 31, 32

  7. [15]

    An improved bound for Sullivan’s convex hull theorem,

    M. Bridgeman, R. Canary and A.Yarmola, “An improved bound for Sullivan’s convex hull theorem,”Proc. LM.S.12(2016), 146–168. 2, 4, 5, 25, 26, 27, 28

    2016

  8. [16]

    Proper affine deformations of positive representations,

    J.-P. Burelle and N. ˇZager Korenjak, “Proper affine deformations of positive representations,” preprint, arXiv:2405.14658. 6

    arXiv

  9. [17]

    Growth of quadratic forms under Anosov subgroups,

    L. Carvajales, “Growth of quadratic forms under Anosov subgroups,”Int. Math. Res. Not. IMRN(2023), 785–854. 21

    2023

  10. [18]

    ‘Proper affine deformations of two-generator Fuchsian groups

    V. Charette, T. Drumm, and W. M. Goldman, “‘Proper affine deformations of two-generator Fuchsian groups”,Trans. Gps., 21, 2016, 953–1002. 6

    2016

  11. [19]

    Lifting representations to covering groups,

    M. Culler, “Lifting representations to covering groups,”Adv. Math.59(1986), 64–70. 20

    1986

  12. [20]

    Geometry and topology of complete Lorentz spacetimes of constant curvature,

    J. Danciger, F. Gu´ eritaud and F. Kassel, “Geometry and topology of complete Lorentz spacetimes of constant curvature,” Ann. E.N.S.49(2016), 1—56. 6

    2016

  13. [21]

    Margulis spacetimes via the arc complex,

    J. Danciger, F. Gu´ eritaud and F. Kassel, “Margulis spacetimes via the arc complex,”Invent. Math.204(2016), 133–193. 6

    2016

  14. [22]

    Proper affine actions of right-angled Coxeter groups,

    J. Danciger, F. Gu´ eritaud and F. Kassel, “Proper affine actions of right-angled Coxeter groups,”Duke Math. J.169(2020), 2231–2280. 2, 5, 6

    2020

  15. [23]

    Affine actions with Hitchin linear part

    J. Danciger and T. Zhang,“ Affine actions with Hitchin linear part”,G.A.F.A.29(2019), 1369–1439. 6, 18

    2019

  16. [24]

    Fundamental polyhedra for Margulis space-times

    T. Drumm, “Fundamental polyhedra for Margulis space-times”,Topology31(1992), 677–683. 6

    1992

  17. [25]

    The geometry of crooked planes,

    T. Drumm, and W. Goldman, “The geometry of crooked planes,”Topology,38(1999), 323–351. 6

    1999

  18. [26]

    Convex co-compact hyperbolic manifolds are determined by their pleating lamination,

    B. Dular and J.M. Schlenker, “Convex co-compact hyperbolic manifolds are determined by their pleating lamination,” preprint, arXiv:2403.10090. 3

    arXiv

  19. [27]

    Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces

    D.B.A. Epstein and A. Marden, “Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces” in Analytical and Geometrical Aspects of Hyperbolic Space, Cambridge University Press, 1987, 113–253. 3, 7, 23, 28 BENDING, ENTROPY AND PROPER AFFINE ACTIONS 33

    1987

  20. [28]

    Quasiconformal homeomorphisms and the convex hull boundary,

    D.B.A. Epstein, A. Marden and V. Markovic, “Quasiconformal homeomorphisms and the convex hull boundary,”Ann. Math. 159(2004), 305–336. 4, 25, 27

    2004

  21. [29]

    Complex earthquakes and deformations of the unit disk,

    D.B.A. Epstein, A. Marden and V. Markovic. “Complex earthquakes and deformations of the unit disk,”J. Diff. Geom. 73(2006), 119–166. 28

    2006

  22. [30]

    Three-dimensional affine crystallographic groups,

    D. Fried and W. Goldman, “Three-dimensional affine crystallographic groups,”Adv. in Math.47(1983), 1–49. 5

    1983

  23. [31]

    Anosov representations: domains of discontinuity and applications,

    O. Guichard and A. Wienhard, “Anosov representations: domains of discontinuity and applications,”Invent. Math. 190(2012), 357–438. 16, 17, 19

    2012

  24. [32]

    Deformation of Fuchsian representations and proper affine actions,

    S. Ghosh. “Deformation of Fuchsian representations and proper affine actions,” preprint, arXiv:2312.16655. 4, 17, 18

    arXiv

  25. [33]

    Avatars of Margulis invariants and proper action

    S. Ghosh, “Avatars of Margulis invariants and proper action.”Ann. Inst. Four., to appear, arXiv:1812.03777. 18

    arXiv

  26. [34]

    Affine Anosov representations and proper actions,

    S. Ghosh and N. Treib, “Affine Anosov representations and proper actions,”IMRN16(2023), 14334–14367. 6

    2023

  27. [35]

    Anosov representations, group manifolds and Laplacian spectra,

    S. Ghosh and T. Kobayashi, “Anosov representations, group manifolds and Laplacian spectra,” in preparation. 21

  28. [36]

    The symplectic nature of fundamental groups of surfaces,

    W. Goldman, “The symplectic nature of fundamental groups of surfaces,”’Adv. Math.,54(1984), 200–225. 2, 3, 5, 22

    1984

  29. [37]

    The complex symplectic geometry ofSL(2,C)-characters over surfaces,

    W. Goldman, “The complex symplectic geometry ofSL(2,C)-characters over surfaces,” inProceedings of the International Conference on Algebraic Groups and Arithmetic Mumbai 2001, Narosa Publishing House (2004), 375–407. 3

    2001

  30. [38]

    Proper affine actions and geodesic flows of hyperbolic surfaces,

    W. Goldman, F. Labourie and G. Margulis, “Proper affine actions and geodesic flows of hyperbolic surfaces,”Ann. of Math. 170(2009), 1051–1083. 6

    2009

  31. [39]

    Flat Lorentz 3-manifolds and cocompact Fuchsian groups

    W. Goldman and G.Margulis, “Flat Lorentz 3-manifolds and cocompact Fuchsian groups” inCrystallographic groups and their generalizationsAMS Contemp. Math.262, 135–145. 18

  32. [40]

    Anosov subgroups: Dynamical and geometric characterizations,

    M. Kapovich, B. Leeb, J. Porti, “Anosov subgroups: Dynamical and geometric characterizations,”Eur. Math. J.3(2017), 808–898. 16

    2017

  33. [41]

    Geometric structures and representations of discrete groups,

    F. Kassel, “Geometric structures and representations of discrete groups,” inProc. I.C.M. 2018, Vol. II, 1115–1151. 16

    2018

  34. [42]

    Affine properness criterion for Anosov representations and generalizations,

    F. Kassel and I. Smilga, “Affine properness criterion for Anosov representations and generalizations,” personal communication,

  35. [43]

    Continuity of convex hull boundaries,

    L. Keen and C. Series, “Continuity of convex hull boundaries,”Pac. J. Math.168(1995), 183–206. 29

    1995

  36. [44]

    Proper action on a homogeneous space of reductive type,

    T. Kobayashi “Proper action on a homogeneous space of reductive type,”Math. Ann.,285(1989) 249–263. 5, 20

    1989

  37. [45]

    The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group,

    B. Kostant, “The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group,”Amer. J. Math.,81(1959), 973–1032. 20, 22

    1959

  38. [46]

    The geometry of bending quasi-Fuchsian groups,

    C. Kourouniotis, “The geometry of bending quasi-Fuchsian groups,” inDiscrete Groups and Geometry (Birmingham, 1991) L.M.S. Lecture Note Series vol. 74, 1992, 148–164. 1, 3, 8, 9

    1991

  39. [47]

    On the continuity of bending

    C. Kourouniotis, “On the continuity of bending”Geometry and Topology Monographs1(1998), 317–334. 1, 3, 8, 9

    1998

  40. [48]

    Anosov flows, surface groups and curves in projective space,

    F. Labourie, “Anosov flows, surface groups and curves in projective space,”Invent. Math.165, 2026, 51–114. 16

    2026

  41. [49]

    Entropy and affine actions for surface groups,

    F. Labourie, “Entropy and affine actions for surface groups,”J. Top.15(2022), 1017–1033. 6

    2022

  42. [50]

    Lehto,Univalent Functions and Teichm¨ uller Spaces, Springer-Verlag, 1987

    O. Lehto,Univalent Functions and Teichm¨ uller Spaces, Springer-Verlag, 1987. 31

    1987

  43. [51]

    Free properly discontinuous groups of affine transformations,

    G. Margulis, “Free properly discontinuous groups of affine transformations,”Dokl. Akad.Nauk SSSR272(1983), 785–788. 5, 14

    1983

  44. [52]

    Complete affine locally flat manifolds with a free fundamental group

    G. Margulis, “Complete affine locally flat manifolds with a free fundamental group”,J. Soviet Math.134(1987), 129–134 5

    1987

  45. [53]

    Lorentz spacetimes of constant curvature,

    G. Mess, “Lorentz spacetimes of constant curvature,”Geom. Ded.,126(2007), 3–45 6

    2007

  46. [54]

    On fundamental groups of complete affinely flat manifolds,

    J.Milnor, “On fundamental groups of complete affinely flat manifolds,”Adv. Math.25(1977), 178–187. 5

    1977

  47. [55]

    The Schwarzian derivative and schlicht functions,

    Z. Nehari, “The Schwarzian derivative and schlicht functions,”Bull. A.M.S.55(1949), 545–551. 30

    1949

  48. [56]

    Complex symplectic geometry of quasi-Fuchsian Space,

    I. Platis,“ Complex symplectic geometry of quasi-Fuchsian Space,”Geom. Ded.87(2001), 17–34 3

    2001

  49. [57]

    Repellers for real analytic maps,

    D. Ruelle, “Repellers for real analytic maps,”Erg. Thy. Dyn. Sys.2(1982), 99–107. 1, 2

    1982

  50. [58]

    A report on an ergodic dichotomy,

    A. Sambarino, “A report on an ergodic dichotomy,”Erg. Thy. Dyn. Sys.44(2024), 236–289

    2024

  51. [59]

    Asymptotic properties of infinitesimal characters and applications,

    A. Sambarino, “Asymptotic properties of infinitesimal characters and applications,” preprint, arXiv:2406.06250. 4, 5, 10, 18, 19, 20

  52. [60]

    An extension of Wolpert’s derivative formula,

    C. Series, “An extension of Wolpert’s derivative formula,”Pac. J. Math.197(2001), 223–239. 8

    2001

  53. [61]

    Proper affine actions on semisimple Lie algebras

    I. Smilga, “Proper affine actions on semisimple Lie algebras.”Ann. Inst. Four.66(2016), 785–831. 6, 14, 17, 19

    2016

  54. [62]

    Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups,

    D. Sullivan, “Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups,”Acta Math. 153(1984), 259–277. 2

    1984

  55. [63]

    Thurston,The Geometry and Topology of 3-manifolds, lecture notes, Princeton University, 1980

    W. Thurston,The Geometry and Topology of 3-manifolds, lecture notes, Princeton University, 1980. 7

    1980

  56. [64]

    Properly discontinuous group actions on affine homogeneous spaces,

    G. Tomanov, “Properly discontinuous group actions on affine homogeneous spaces,”Proc. Steklov Inst. Math.292(2016), 260–271. 5

    2016

  57. [65]

    An invitation to higher Teichm¨ uller theory,

    A. Wienhard, “An invitation to higher Teichm¨ uller theory,” inProc. I.C.M. 2018vol. II, 1013–1039. 16

  58. [66]

    Constructing proper affine actions via higher strip deformations,

    N. ˇZager Korenjak, “Constructing proper affine actions via higher strip deformations,” preprint, arXiv:2205.14712. 6

    arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.