Spherical CR uniformizations of a sequence of hyperbolic 3-manifolds

Baohua Xie; Jiming Ma

REVIEW 4 major objections 5 minor 28 references

Spherical CR uniformizations of a sequence of hyperbolic 3-manifolds

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves that for each integer $n \geq 5$, the hyperbolic 3-manifold obtained by Dehn filling the census manifold $s782$ along the slope $(n-1)m_1 + l_1$ admits a spherical CR uniformization.

desk verdict A second explicit infinite family of CR-uniformizable Dehn fillings, built on a credible strategy but with real proof gaps that need repair before I'd rely on the full theorem. read the letter →

arxiv 2608.11571 v1 pith:4NPTVMF2 submitted 2026-08-12 math.GT

Jiming Ma , Baohua Xie This is my paper · ORCID

classification math.GT MSC 20H1057M5022E4051M10

keywords complexhyperbolicgeometrysphericalCRuniformizationtrianglegroupscusped3-manifoldsDehnfillingsDirichletdomainFord

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 each integer $n\geq 5$, the hyperbolic 3-manifold obtained from the two-cusped census manifold $s782$ by Dehn filling its second cusp along the slope $(n-1)m_1+l_1$ admits a spherical CR uniformization—a realization as the quotient of part of the boundary of complex hyperbolic space by a discrete group. The uniformizing group is explicit: it is the even subgroup of the complex hyperbolic triangle group $\Delta_{4,4,n;\infty}$, and the filled manifold is exactly its 3-manifold at infinity. This matters because explicit infinite families of spherical CR-uniformizable hyperbolic 3-manifolds are very rare; previously the Whitehead link complement and its Dehn fillings were essentially the only concrete case. The proof compares the Ford domain of $\Delta_{4,4,\infty;\infty}$ with the Dirichlet domain of $\Delta_{4,4,n;\infty}$, shows they share the same side combinatorics and topology, and reads the Dehn filling slope off a boundary curve that bounds an essential disk.

What carries the argument

The load-bearing machinery is a side-by-side comparison of two polyhedra in the complex hyperbolic plane: the Ford domain $F$ of the even subgroup of $\Delta_{4,4,\infty;\infty}$ and the Dirichlet domain $D$ of the even subgroup of $\Delta_{4,4,n;\infty}$, centered at the fixed point $o$ of $I_1I_2$. Each is cut out by bisectors—hypersurfaces of points equidistant from $o$ and a group translate—tagged $B_k^+,B_k^-,B_k^\star,B_k^\diamond$ for $k$ modulo $n$ (or $k\in\mathbb{Z}$ in the $\infty$ case), and the core of the proof is a long verification that these bisectors intersect in exactly the same pattern. That verification uses trace bounds and the disjointness criterion for cyclic groups (Proposition 5.9, from [20]), balanced-pair technology for triple intersections, and the Poincaré polyhedron theorem to conclude that $D$ is a Dirichlet domain with presentation $\langle S,A\mid A^n=S^4=(AS)^4=1\rangle$. The quotient of $D_\infty$ by the $\mathbb{Z}_n$-action is then a solid torus whose boundary curve $C_n$ encodes the Dehn filling slope.

What would settle it

Compute, with rigorous interval arithmetic, the sign of the polynomial $F(c,c_m)$ in Proposition 5.11 for $n=7$ and $m=3$; if it is negative, the bisectors $B_0^+$ and $B_m^+$ intersect and the Dirichlet domain pattern differs from the paper's claim. Alternatively, build the quotient of $D_\infty$ for $n=7$ and compare its fundamental group with that of the Dehn filling of $s782$ along $6m_1+l_1$; a mismatch would refute Theorem 1.1.

Watch

Extended reading notes

Core claim

The central theorem (Theorem 1.1) asserts that the representation $\rho: T_{4,4,n}\to \mathrm{PU}(2,1)$ with image $\Delta_{4,4,n;\infty}$ is a discrete embedding for every $n\geq5$, and that the 3-manifold at infinity $M_{4,4,n;\infty}$ of the even subgroup $\langle I_1I_2,I_2I_3\rangle$ is precisely the 1-cusped hyperbolic manifold obtained from $s782$ by Dehn filling the second cusp along $(n-1)m_1+l_1$. The discovery is carried by a structural comparison: the Dirichlet domain $D$ of the even subgroup of $\Delta_{4,4,n;\infty}$ has the same local combinatorial pattern of ridges, sides, and side-pairings as the Ford domain $F$ of the even subgroup of $\Delta_{4,4,\infty;\infty}$, with the only global change being that the $\mathbb{Z}$-action of $I_1I_2$ on $F_\infty$ is replaced by a $\mathbb{Z}_n$-action on $D_\infty$. That change turns the end $T^2\times[1,\infty)$ of the quotient into a solid torus $D^2\times S^1$, exactly a Dehn filling, and the slope is identified by showing that a boundary curve $C_n$ of length $3n+1$ bounds an essential disk and corresponds to $l_1+(n-1)m_1$ in the meridian-longitude system of the second cusp of $s782$.

Load-bearing premise

The construction rests on the claim that certain large polynomial expressions built from the angle $\pi/n$ remain positive for every $n$ and every pair of sides, and the paper verifies the decisive cases with a combination of analytic estimates, numerical checks, and graphical inspection rather than a complete rigorous certificate; if even one of those sign assertions fails, the Dirichlet domain's side pattern differs from the Ford domain's and the Dehn filling conclusion collapses.

Editorial extensions

If this is right

Each $M_{4,4,n;\infty}$ is simultaneously a hyperbolic 3-manifold (as a Dehn filling of $s782$) and a spherical CR-uniformizable 3-manifold (as the 3-manifold at infinity of a discrete group), so uniformizable spherical CR structures occur for an explicit infinite family of hyperbolic manifolds.
The discreteness statement confirms the relevant case of Schwartz's conjecture on complex hyperbolic triangle groups, giving new discrete embeddings of triangle groups into $\mathrm{PU}(2,1)$ for all $n\geq5$.
The slope formula is explicit and effectively computable, so each filled manifold is exhibited with its uniformizing group rather than merely shown to exist.
The $n=5$ case receives an independent check from a direct fundamental-group computation of the side-pairing, matching the same Dehn filling and anchoring the general slope formula.

Reading between the lines

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

A natural reading of the construction is as a complex hyperbolic analogue of Dehn surgery: as $n$ grows, the Dirichlet domain $D_n$ ought to converge to the Ford domain $F$, with the filled solid torus shrinking to the cusp; making that convergence explicit would connect the algebra to the classical geometric picture of Dehn surgery.
A rigorous interval-arithmetic certificate for the sign of the polynomial $F(c,c_m)$ in Propositions 5.11 and 5.12 would close the only gap in the proof and would make the theorem independently checkable for any chosen $n$.
The same Ford-domain-versus-Dirichlet-domain comparison may work for other two-cusped manifolds uniformized by complex hyperbolic triangle groups; if so, each would yield another explicit infinite family of spherical CR Dehn fillings, beyond the Whitehead link complement and $s782$.
The slope-detection method centered on the boundary curve $C_n$ suggests a general recipe: in domain comparisons of this type, the Dehn filling slope can be read from a distinguished boundary curve whose length grows linearly with the group parameter.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline: this is the second explicit infinite family of Dehn fillings of a cusped hyperbolic 3-manifold with spherical CR uniformization, after Acosta's Whitehead link complement. The authors prove that for each n≥5, filling s782 along (n−1)m1+l1 works, by comparing the Ford domain of Δ_{4,4,∞;∞} with the Dirichlet domain of Δ_{4,4,n;∞}. That family is new, and the Dirichlet domain analysis is a substantial computation.

What's genuinely good: the strategy is credible and follows the Schwartz/Acosta template without being a rerun. The n=5 case gets an independent SnapPy check, which is a real sanity check. The paper is also honest about its own gaps: it explicitly says n=6 is omitted from the Section 5 proof, and it openly defers Proposition 6.1 to Acosta's paper. The reliance on [10] for the base manifold is legitimate; the new content is the comparison and the slope, not the base identification.

Now the soft spots, in proportion. Proposition 6.1 is load-bearing: it identifies the Dehn filling slope, and the proof is only 'similar to' Acosta. That's a deferral, not a proof. The n=6 exclusion means Theorem 1.1 as stated is not actually proved for all n≥5. More concerning, the comparison of Dirichlet and Ford domains rests on Propositions 5.11 and 5.12, where non-intersection of bisectors reduces to polynomial inequalities. The text says these can be verified 'using a combination of analytical and numerical methods' and that 'standard techniques from elementary calculus confirm' non-negativity, but no certificate or complete calculation is supplied. If one of those inequalities fails for some n or m, the side pattern could change and the manifold at infinity need not be the stated filling. That's not a manufactured flaw; the authors flag it themselves by appealing to numerics and graphics.

Who this is for: people working on spherical CR structures and complex hyperbolic triangle groups. The n=5 verification and the explicit family are valuable even if the general proof needs repair.

My recommendation: this deserves a serious referee and should not be desk-rejected. But I would not accept it as-is. The authors should either complete the polynomial inequalities with rigorous certificates (interval arithmetic or exact algebraic verification) or restrict the theorem to n≥7 plus the separately checked n=5, and they need a real proof of Proposition 6.1, not a citation. Once those are done, this is a solid result.

Referee Report

4 major / 5 minor

Summary. The paper proves that for each integer n ≥ 5, the one-cusped hyperbolic 3-manifold obtained from the SnapPy census manifold s782 by Dehn filling on the second cusp along the slope (n−1)m₁ + l₁ admits a spherical CR uniformization. The proof compares the Ford domain of the even subgroup of the complex hyperbolic triangle group Δ_{4,4,∞;∞} with the Dirichlet domain of the even subgroup of Δ_{4,4,n;∞}, showing that the two domains have the same local combinatorics and topology. A separate computation identifies the n = 5 case directly with the corresponding SnapPy Dehn filling, and the paper frames the result as an explicit infinite family of spherical CR uniformizations, analogous to Acosta's Whitehead link construction and confirming a special case of a conjecture of Schwartz.

Significance. If the proof is completed, the result would be a significant and explicit contribution to spherical CR geometry: it would provide the second known infinite family of Dehn fillings of a concrete cusped hyperbolic 3-manifold that admit uniformizable spherical CR structures, and it would give a complex-hyperbolic analogue of Thurston's hyperbolic Dehn surgery theorem for this family. The paper has clear strengths: the main strategy is well motivated, the n = 5 case is checked against SnapPy with a computed presentation, the base case n = ∞ is re-derived with adjusted labels, and the authors are explicit about where their arguments rely on prior work. These strengths are considerable; however, the central proof currently has load-bearing gaps in the verification of the bisector intersection pattern, the omitted n = 6 case, and the deferred proof of the Dehn-filling slope.

major comments (4)

[§5.4, Proposition 5.11 and Proposition 5.12] The proof that the Dirichlet domain D has the same side combinatorics as the Ford domain F reduces to the non-intersection of bisectors, which in turn reduces to sign claims on explicit polynomials. For Proposition 5.11, the proof ends with the statement that nonnegativity of the relevant polynomial 'can be verified using a combination of analytical and numerical methods' and that 'graphical verification further supports this conclusion'; no rigorous certificate, exact algebraic proof, or reproducible code is supplied. Proposition 5.12 similarly ends with 'Standard techniques from elementary calculus confirm its non-negativity' without giving the calculation. These inequalities are load-bearing: if any of them failed for some n or m, the bisectors could meet in a different pattern, the side pairing of D would differ from F, and the manifold at infinity would not be the stated Dehn filling. A complete proof must supply a rigorous verification, for example a Sturm sequence or cylindrical algebraic decomposition argument, or interval arithmetic with formal error bounds.
[§5, first paragraph after Definition 5.4] Theorem 5.5 is stated for n ≥ 5, but the proof in Section 5 is carried out only for n ≥ 7. The text says the cases n = 5 and n = 6 are 'slightly simpler' and their proofs are omitted. However, Theorem 1.1 claims all n ≥ 5. The n = 5 case is checked separately in Section 7, but the n = 6 case is not treated anywhere in the paper. Since the claimed theorem is an infinite family, the n = 6 case must either be proved explicitly or the statement of the theorem must be restricted to n ≥ 7 (with n = 5 handled separately).
[§6, Proposition 6.1] Proposition 6.1 states that the curve C_n on the bounding torus of D∞ bounds an essential disk in D∞, and the proof is deferred with the single sentence 'The proof is similar to the first part of Proposition 7.8 in [1].' This proposition is essential for identifying the Dehn filling slope in Theorem 1.1(ii): without it, the curve that bounds the disk in the solid torus W_D is not determined, and the slope (n−1)m₁ + l₁ does not follow. A concise but complete proof must be included in the paper, rather than referenced to a different context with a different group and different notation.
[§7, Proposition 7.1] The direct verification for n = 5 relies on a 2-cell decomposition of the boundary of a 3-ball N, but the text states that the vertex identifications are 'recorded on our scratch paper' and are omitted. The edge-cycle relations in Table 7 are given, and the resulting presentation is checked against SnapPy, so the logical structure is visible. Nevertheless, for a paper whose novelty is an explicit infinite family, the omitted bookkeeping makes independent verification unnecessarily difficult. The authors should either include the vertex identification data in an appendix or provide the computer code used to assemble the cycles, so that the n = 5 verification is reproducible.

minor comments (5)

[§5.4, Proposition 5.11(i) vs (iii)] The sign conventions for the function F(c, c_m) are confusing: in Item (i) the text says 'We aim to prove that F(c, c_m) > 0', while in Item (iii) it says 'Our goal is to prove that F(c, c_m) < 0' before displaying an expression for −F. The two statements should be separated with explicit notation, e.g., F₁ and F₃, to avoid ambiguity.
[§5.1, text after Proposition 5.1] The sentence 'By a straightforward verification, I₁, I₂ and I₃ are complex reflections in PU(2,1) satisfying the required group relations' is acceptable, but the text should state explicitly that the parameters a and b are real and that the matrix entries are compatible with the Hermitian form; otherwise the reader cannot check that the matrices preserve the form without redoing the omitted computation from [15].
[§6, paragraph after Proposition 6.1] The phrase 'this curve corresponds to the slope l₁ + (n−1)m₁' is written as 'f₂f_n¹' in the preceding sentence, which is a typographical corruption of f₂ f₁ⁿ; this should be corrected to the standard notation f₂ f₁ⁿ.
[§7, paragraph after Figure 9] The sentence 'Since D∞ covers T_D, we use s(S) ∩ T_D in T_D to denote the image of s(S) ∩ D∞ ∩ A to simplify notation' contains a typo ('usess') and is hard to parse; rewriting it would improve clarity.
[§5.6, Theorem 5.26] Theorem 5.26 is stated as a corollary but its proof is distributed through the preceding propositions and the Poincaré polyhedron theorem application; it would help to spell out explicitly which propositions establish the claimed equality of the 2-cell decompositions on the boundaries, since this theorem is the bridge to the Dehn filling conclusion in Section 6.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the new Dehn-filling family is obtained from an explicit comparison of the Ford domain of Δ4,4,∞;∞ with the Dirichlet domain of Δ4,4,n;∞, not from fitting or assuming the conclusion.

full rationale

The base identification M4,4,∞;∞ ≅ s782 is taken from the authors' earlier paper [10], and the present paper re-derives the isomorphism π1(M4,4,∞;∞) ≅ π1(s782) with adjusted labels; this is an input, not the target conclusion. The genuinely new content is Theorem 5.5, which asserts that for n ≥ 5 the Dirichlet domain D of the even subgroup of Δ4,4,n;∞ exists with the stated side combinatorics. The proof is a direct bisector computation for the explicit matrices in Proposition 5.1, with c = cos(π/n) fixed by requiring I1I2 to have order n; no parameter is fitted to the Dehn-filling slope. The slope (n−1)m1 + l1 is derived in Section 6 from the curve Cn and the identification f1 = c^{-1}a, f2 = bc, and Section 7 independently checks n = 5 against SnapPy's 4m1 + l1 filling. Citations [10] and [15] are self-citations, but they supply the base-domain combinatorics and the matrix parameter ansatz respectively; both are explicit and are not equivalent to the theorem being proved. The manuscript does contain gaps—Theorem 5.5 is proved only for n ≥ 7 ('Their proofs are slightly simpler, so we omit their proof herein'), and Propositions 5.11 and 5.12 rely on unstated analytical/numerical verification ('can be verified using a combination of analytical and numerical methods... Graphical verification further supports this conclusion')—but these are completeness and rigor gaps, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard results in complex hyperbolic geometry, prior published work on the Ford domain of Δ_{4,4,∞;∞} and its identification with s782, SnapPy census data, and Magma simplifications. No new entities are postulated and no data-fitting parameters are used. The main unproved analytical input is the sign of certain polynomials, recorded here as an ad hoc assumption.

assumptions (7)

standard math Poincaré polyhedron theorem (as formulated in [6])
Used in Section 5.6 to conclude D is a Dirichlet domain and Σ is discrete with the given presentation.
standard math Phillips's theorem on Dirichlet polyhedra for cyclic groups (Prop 5.9)
Used in Propositions 5.11 and 5.12 to conclude certain bisectors are disjoint when traces exceed 3.
standard math Deraux-Parker-Paupert Lemma 2.3 (Prop 5.10)
Used in Proposition 5.17 to identify intersections of bisectors with complex lines as real geodesics.
domain assumption Ford domain combinatorics of Δ_{4,4,∞;∞} and identification of M_{4,4,∞;∞} with s782 from [10]
Section 4 takes the results of Jiang-Wang-Xie as given; the comparison in Theorem 5.26 and the base manifold identification depend on them.
domain assumption SnapPy census data (s782, its meridian-longitude system, and the filled manifold L)
The cusp labeling and the n=5 verification in Section 7 use SnapPy as ground truth.
domain assumption Magma simplifications of group presentations
Sections 4.2 and 7 rely on Magma to simplify presentations; scripts are not provided.
ad hoc to paper Sign of the polynomials F(c,cm), Γ_{2,1}, Γ_{2,2} and related expressions on the stated domains
Stated in Proposition 5.11 and 5.12 proofs with numerical methods and graphical verification instead of a complete proof; the entire non-intersection pattern depends on them.

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Pith. "Pith review of Spherical CR uniformizations of a sequence of hyperbolic 3-manifolds." pith.science (2026). https://pith.science/paper/4NPTVMF2

@misc{pith2026260811571,
  author       = {Pith},
  title        = {Pith review of: Spherical CR uniformizations of a sequence of hyperbolic 3-manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4NPTVMF2}},
  note         = {Machine review of arXiv:2608.11571}
}

abstract

Let $s782$ be the 2-cusped hyperbolic 3-manifold in the SnapPy census. Its spherical CR uniformization was established in \cite{JWX2023} using the Ford domain of the complex hyperbolic triangle group $\Delta_{4,4,\infty;\infty}$. By comparing the combinatorial structures of the Ford domain of $\Delta_{4,4,\infty;\infty}$ and the Dirichlet domain of $\Delta_{4,4,n;\infty}$, we prove that for each $n \geq 5$, the Dehn filling of $s782$ along the slope $(n-1)\mathcal{m}_1 + \mathcal{l}_1$ on its second cusp admits a spherical CR uniformization, where $(\mathcal{m}_1, \mathcal{l}_1)$ denotes the meridian-longitude system of a cusp in SnapPy notation.

Figures

Figures reproduced from arXiv: 2608.11571 by the authors.

**Figure 2.** Schematic views of the 2-cell decomposition of ∂s + 0 (left) and ∂s − 0 (right). On the left, we use s ∗ 0 to represents s + 0 ∩ s ∗ 0 . The other labels obey the same rule. u2 y1 v1 x1 w1 u1 t2 s2 p2 r2 q2 t1 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** A schematic view of the 3-sides s ⋆ 0 (left) and s ⋄ 0 (right). The shadowed regions are ∂∞s ⋆ 0 and ∂∞s ⋄ 0 , respectively. To use the Poincaré polyhedron theorem for the Ford domain F, we consider the following: The side-pairing maps: A kSA−k : s − k 7→ s + k , AkS 2A −k : s ⋆ k 7→ s ⋆ k , AkQA−k : s ⋄ k 7→ s ⋄ k The ridge circles: (s − k ∩ s + k , s − k , s + k ) AkSA−k −−−−−→ (s ⋆ k ∩ s + k , s ⋆ k , s + k ) AkS… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: A fundamental domain of the Z = ⟨A⟩-action on F∞ with a cutting disk D. u1 v1 w1 x1 y1 u2 u1 v1 w1 x1 y1 u2 s1 r1 q1 p1 p2 q2 r2 s2 t1 S4 S8 S −1 8 S −1 4 R− R+ D+ D− E A(E) Q+ A−1Q−A [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: The 2-cell decomposition of the 2-sphere boundary of N to get the 3-manifold M4,4,∞;∞. The 3-manifold M4,4,∞;∞ is the quotient space of N. The side-pairings on N are f1 : D− −→ D+, f2 = A : E −→ A(E), f3 = R : R+ −→ R−, f4 = A −1Q : Q+ −→ A −1Q−A, f5 = S −1 : S8 −→ S −…

**Figure 6.** Figure 6: A schematic view of the Dirichlet domain of complex triangle group ∆(4, 4, n). Definition 5.4. Define the polyhedron D in H2 C by D = ( p ∈ H2 C [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: An abstract picture of the Dirichlet domain of the even subgroup of ∆4,4,6;∞. Although the proof is presented for n ⩾ 7, the combinatorial picture for n = 6 is qualitatively similar and is shown here for illustration. The red regions are ∂∞s + 0 , ∂∞s + 2 and ∂∞s + 4 ,…

**Figure 8.** Figure 8: A realistic view of the ideal boundary D∞ of the Dirichlet domain D of ∆4,4,8;∞. It is the solid torus outside all of the spheres. Lemma 5.7. Let n1, n2, n3 be the polar vectors of I1, I2, I3, respectively: n1 =   1 − c s 0   , n2 =   c − 1 s 0   , n3 =   a b…

**Figure 9.** Figure 9: A fundamental domain A of the ⟨I1I2⟩-action on the torus ∂D∞, which projects to the torus ∂TD. This provides a 2-cell decomposition of ∂TD together with the boundary curve (in red) of an essential disk in TD. of the triangle labeled ARA−1 ) arise from the intersections…

**Figure 10.** Figure 10: The 2-cell decomposition of the 2-sphere boundary of N to get the 3-manifold M4,4,5;∞. When calculating edge circles, the edge labeled by i is referred to as edge ei . Based on the above analysis and in comparison with the side-pairings on ∂TD, the side-pairings on ∂N…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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W. M. Goldman and J. R. Parker

work page

[2] [1]

M. Acosta . Spherical CR uniformization of Dehn surgeries of the Whitehead link complement. Geom. Topol. 23 (2019), 2593--2664

work page 2019

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M. Culler, N. M. Dunfield, M. Goerner, and J. R. Weeks . SnapPy, a computer program for studying the geometry and topology of 3-manifolds. http://snappy.computop.org

work page

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M. Deraux and E. Falbel . Complex hyperbolic geometry of the figure eight knot. Geom. Topol. 19 (2015), 237--293

work page 2015

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work page 2015

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M. Deraux . A 1 -parameter family of spherical CR uniformizations of the figure eight knot complement. Geom. Topol. 20 (2016), 3517--3621

work page 2016

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M. Deraux, J. R. Parker and J. Paupert . New non-arithmetic complex hyperbolic lattices. Invent. Math. 203 (2016), 681--771

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E. Falbel, A. Guilloux and P. Will

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