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REVIEW 4 major objections 4 minor 33 references

Nil 3-manifolds and cusps of complex hyperbolic surfaces

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The seven families of compact Nil 3-manifolds are completely classified by which commensurability class of complex hyperbolic 2-manifolds they can bound as cusp cross-sections.

desk verdict The arithmetic classification in Theorems 1–3 is solid and worth refereeing; the proof of Theorem 4 has a real gap for the antiholomorphic families (3) and (4), and the abstract overstates the non-arithmetic conclusion. read the letter →

arxiv 2411.15345 v1 pith:XT3HBUWV submitted 2024-11-22 math.GT math.DGmath.GR

Julien Paupert , Connor Sell This is my paper · ORCID
classification math.GTmath.DGmath.GR MSC 57K2032Q4522E4011F06
keywords Nil3-manifoldscomplexhyperbolicsurfacescuspcross-sectionsPicardmodulargroupscommensurabilityclassesHeisenberggroupnon-arithmeticlatticesholonomyrepresentations
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 settles, for compact Nil 3-manifolds, which commensurability classes of cusped arithmetic complex hyperbolic 2-manifolds realize a given manifold as a cusp cross-section. Four of the seven Nil-manifold families (Nil-tori, vertical and horizontal half-twists, double half-twists) occur in the commensurability class of every Picard modular group $\mathrm{PU}(2,1,\mathcal{O}_d)$, while the remaining three families (1/4-twist, 1/3-twist, 1/6-twist) each occur in exactly one class, pinned to $d=1$ or $d=3$. The paper also proves that every compact Nil 3-manifold occurs as a cusp cross-section of some non-arithmetic complex hyperbolic 2-manifold or $\mathbb{H}^2_{\mathbb{C}}$-manifold, and that the horizontal half-twist and double half-twist families occur only as $\mathbb{H}^2_{\mathbb{C}}$-manifolds, never as complex hyperbolic manifolds. Together these results give a complete answer to the geometric boundary problem for complex hyperbolic surfaces in real dimension three.

What carries the argument

The load-bearing object is the 3-dimensional Heisenberg group $\mathrm{Nil}$, viewed as the punctured boundary of complex hyperbolic 2-space, with isometry group $\mathrm{Isom}(\mathrm{Nil}) = \mathrm{Nil} \rtimes (\mathrm{U}(1) \rtimes \mathbb{Z}/2)$. The paper writes explicit holonomy representations of the seven Nil-manifold group presentations into the cusp subgroups $G_\infty(\mathcal{O}_d)$ of the Picard modular groups, checking the presentation relations with the group law of $\mathrm{Nil}$. The obstruction part uses the rotational part $u$ of the ellipto-parabolic generator $\alpha$: for families (5)-(7), $u$ is $\zeta_4$, $\zeta_3$, or $\zeta_6$, and because a matrix representative in $\mathrm{U}(2,1,E_d)$ has eigenvalues $1,1,u$ lying in a cubic extension of the field of definition $E_d$, the only possible $d$ are $1$ or $3$. For the non-arithmetic existence, the paper uses the fact that the known non-arithmetic lattices have cusp groups isomorphic to $G_\infty(\mathcal{O}_1)$ or $G_\infty(\mathcal{O}_3)$ (from [DPP2] and [DFP]), and then applies the separability lemma in [B] together with the covering arguments of [M1] and [M2] to promote an immersed cusp to an embedded one in a torsion-free finite cover.

What would settle it

The most direct check is to search for a holonomy representation of a 1/4-twist Nil 3-manifold into $G_\infty(\mathcal{O}_d)$ for a squarefree $d\neq 1$ with the required rotational part $\zeta_4$; the paper's eigenvalue argument says the eigenvalues $1,1,\zeta_4$ would have to lie in a cubic extension of $\mathbb{Q}(i\sqrt{d})$, which is impossible, so any such representation would refute Theorem 2.

Watch

Extended reading notes

Core claim

The central discovery is that the seven families of compact Nil 3-manifolds split into universal and rigid cusp shapes for arithmetic complex hyperbolic surfaces. Using explicit holonomy representations into the cusp subgroups $G_\infty(\mathcal{O}_d)$ of $\mathrm{Isom}(\mathrm{Nil})$, the paper shows in Theorem 1 that families (1)–(4) appear as cusp cross-sections in the commensurability class of $\mathrm{PU}(2,1,\mathcal{O}_d)$ for every squarefree $d$, meaning these shapes occur in every arithmetic commensurability class. In Theorem 2, families (5)–(7) are shown to occur in only one commensurability class each: the 1/4-twist only for $d=1$ and the 1/3- and 1/6-twists only for $d=3$, via an eigenvalue argument that forces the rotational part $\zeta_4$ or $\zeta_3/\zeta_6$ to lie in the field of definition of the class. Theorem 3 shows that the horizontal half-twist and double half-twist families can never be cusp cross-sections of complex hyperbolic manifolds because every holonomy representation must contain antiholomorphic isometries. Finally, Theorem 4 combines the explicit representations with the cusp groups of the known non-arithmetic lattices to prove that every compact Nil 3-manifold occurs as a cusp cross-section of some non-arithmetic complex hyperbolic 2-manifold or $\mathbb{H}^2_{\mathbb{C}}$-manifold.

Load-bearing premise

The proof of the non-arithmetic existence theorem (Theorem 4) assumes that the separability-based promotion of an immersed cusp to an embedded cusp cross-section, known for arithmetic lattices, also works for the specific non-arithmetic lattices used here, including those whose cusp stabilizers contain antiholomorphic isometries.

Editorial extensions

If this is right

  • For every squarefree $d$, the Picard modular group $\mathrm{PU}(2,1,\mathcal{O}_d)$ has cusps whose cross-sections are homeomorphic to Nil-tori, vertical half-twists, horizontal half-twists, and double half-twists.
  • The 1/4-twist Nil-manifold can appear as a cusp cross-section only in the commensurability class of $\mathrm{PU}(2,1,\mathcal{O}_1)$, and the 1/3- and 1/6-twist manifolds only in the class of $\mathrm{PU}(2,1,\mathcal{O}_3)$.
  • Families (3) and (4) cannot be cusp cross-sections of complex hyperbolic manifolds; any quotient with such a cusp must be an $\mathbb{H}^2_{\mathbb{C}}$-manifold whose holonomy contains antiholomorphic isometries.
  • Every compact Nil 3-manifold is realized as a cusp cross-section of some non-arithmetic complex hyperbolic 2-manifold or $\mathbb{H}^2_{\mathbb{C}}$-manifold, extending the arithmetic existence result of [M1] and [M2] to non-arithmetic settings.
  • The classification is exhaustive: each of the seven Nil-manifold families is either universal (occurring in all arithmetic commensurability classes) or rigid (occurring in exactly one).

Reading between the lines

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

  • The eigenvalue obstruction that pins families (5)-(7) to $d=1$ or $d=3$ suggests a general constraint for almost-flat manifolds modelled on generalized Heisenberg groups: the rotational part of a cusp stabilizer must lie in a field compatible with the lattice's field of definition, which could constrain higher-dimensional complex and quaternion hyperbolic cusps.
  • The non-arithmetic existence proof likely generalizes to any non-arithmetic lattice that contains a cusp group isomorphic to $G_\infty(\mathcal{O}_1)$ or $G_\infty(\mathcal{O}_3)$ and satisfies the integrality condition used here, so the result may apply beyond the known examples.
  • A natural next question is which Nil-tori occur as cusp cross-sections of one-cusped complex hyperbolic 2-manifolds; the explicit representations here may help determine the possible Euler numbers, since the one-cusped case is restricted to Nil-tori with Euler number a multiple of 4.
  • The complete classification for complex hyperbolic surfaces provides a template for the analogous question for quaternion hyperbolic manifolds, where the almost-flat cusp cross-sections are modelled on higher-dimensional Heisenberg groups and no comparable classification is currently known.
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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 / 4 minor

Summary. The paper studies which compact Nil 3-manifolds occur as cusp cross-sections of complex hyperbolic 2-manifolds and of the more general H^2_C-manifolds, with focus on the arithmetic commensurability classes determined by the Picard modular groups PU(2,1,O_d). The authors give explicit holonomy representations for all seven families of Nil 3-manifolds: families (1)--(4) are embedded into G∞(O_d) for every squarefree d, and families (5)--(7) are shown by an eigenvalue obstruction to occur only for d=1 or d=3. Theorem 3 establishes that families (3) and (4) cannot appear in any holomorphic quotient and must have antiholomorphic cusp holonomy. The paper also claims, in Theorem 4, that every Nil 3-manifold occurs as a cusp cross-section of a non-arithmetic complex hyperbolic or H^2_C-manifold. The arithmetic part is supported by explicit, checkable matrix computations; the non-arithmetic part has a gap in the promotion argument for the antiholomorphic families.

Significance. The arithmetic classification in Theorems 1--3 is a substantial and largely convincing contribution: it answers, for Nil 3-manifolds, the analogue of Sell's classification for flat 3-manifolds, and it supplies corrected explicit holonomy representations, including a repair of an incorrect representation in the appendix of [M1] for the double half-twist family. The eigenvalue obstruction in Theorem 2 is clean and gives a concrete falsifiable criterion. The holomorphic/antiholomorphic dichotomy in Theorem 3 is also useful. If the non-arithmetic claim in Theorem 4 can be repaired, it would be a significant extension of McReynolds' existence theorem; as written, however, the proof does not establish the claimed result for families (3) and (4), and the abstract overstates the theorem.

major comments (4)
  1. [Abstract] The abstract states: "We also show that every compact Nil 3-manifold occurs as the cusp cross-section of some non-arithmetic complex hyperbolic 2-manifold." This is inconsistent with Theorem 3, which says that families (3) and (4) are not cusp cross-sections of any complex hyperbolic manifold or orbifold and can occur only in H^2_C-manifolds. The abstract must be corrected to "complex hyperbolic 2-manifold or H^2_C-manifold", or the statement of Theorem 4 must be revised so that the announced claim matches what is actually proved.
  2. [§3, Theorem 1 and Lemma 5] There is an unresolved tension between Theorem 1 and Theorem 3 for families (3) and (4). Lemma 5 constructs holonomy representations for these families into G∞(O_d) = G^0∞(O_d) ∪ G^0∞(O_d)σ, so the cusp group contains antiholomorphic isometries, and the resulting quotient is an H^2_C-manifold rather than a complex hyperbolic manifold. If "commensurability class of PU(2,1,O_d)" means commensurability among torsion-free subgroups of PU(2,1), then Theorem 1 is false for families (3) and (4) by Theorem 3. If it means commensurability in the full isometry group Isom(H^2_C), this must be stated explicitly, and the abstract and Section 2.3's identification of commensurability classes with Picard modular groups must be qualified, since that identification is only for holomorphic quotients.
  3. [§3.2, proof of Theorem 4, Step (2)] The promotion argument uses Proposition 2 with G = SU(2,1) and H = Stab_G(q∞). This applies only to lattices in the holomorphic isometry group. For families (3) and (4), the image of ρ2(π1(N)) contains antiholomorphic isometries by Theorem 3 and Lemma 5, so the relevant cusp subgroup lies in G∞(O_d), inside an R-reflection-generated lattice in Isom(H^2_C), not in a lattice in SU(2,1). The conclusion "We now have a non-arithmetic lattice Γ_1 in SU(2,1) such that N is diffeomorphic to a cusp cross-section" cannot hold for families (3) and (4). The paper needs a separate separability argument for the full cusp stabilizer H = Stab_Isom(H^2_C)(q∞), or for the non-holomorphic R-reflection groups ~Γ and ~S, and none is supplied.
  4. [§3.2, proof of Theorem 4, final paragraph] The final step of the promotion argument is not correctly stated. The diagonal embedding φ(Γ_1) is not a subgroup of the first factor of Λ, so "the intersection with the first factor of Λ" cannot contain the diagonal image of ρ3(π1(N)). The intended construction is presumably Γ_2 = φ^{-1}(Λ_0 ∩ φ(Γ_1)) (or an explicit projection followed by intersection with Γ_1), and one must then verify that Γ_2 is torsion-free, has finite index in Γ_1, and still has ρ3(π1(N)) as its full cusp subgroup. As written, this step is incomplete.
minor comments (4)
  1. [Equation (1) and Lemma 2] The displayed Heisenberg group law appears to omit the complex conjugation that makes the form alternating; as printed, Im(z1 z2) is symmetric and does not give the standard 2-step nilpotent Heisenberg group. Please check the convention in equation (1) and in Lemma 2(1), since the computations in Section 3 rely on it.
  2. [Theorem 6 statement] The statement contains a duplicated word: "diffeomorphic to to a cusp cross-section" should be "diffeomorphic to a cusp cross-section".
  3. [Theorem 3 statement] The word "anitholomorphic" should be "antiholomorphic".
  4. [Corollary 2 and proof of Theorem 4] The groups ~Γ and ~S are R-reflection groups in Isom(H^2_C), not lattices in PU(2,1). Corollary 2 and the proof of Theorem 4 should explicitly state this and identify the relevant index-2 holomorphic sublattice, since the current wording sometimes refers to them as lattices in PU(2,1).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the arithmetic classification is self-contained, and the non-arithmetic existence proof rests on independent published results rather than on this paper's own conclusions.

full rationale

The central arithmetic results, Theorems 1 and 2, are established inside the paper by explicit holonomy representations (Lemma 5 and the displayed representations in the proof of Theorem 2) and by an eigenvalue obstruction that uses only the field of definition of the commensurability class together with the second generalized Bieberbach theorem. No parameter is fitted and no target conclusion is assumed. Theorem 3 is likewise proved directly from the group presentations and Lemma 2. The non-arithmetic existence result, Theorem 4, invokes prior published work—[M1], [M2], [DPP2], [DFP], and [PW]—for representations, cusp groups of known non-arithmetic lattices, and the separability promotion construction. Although [DPP2] and [DFP] are co-authored by Paupert, those are independent published classifications with stated assumptions, not results defined by this paper, so they are external support rather than self-referential inputs. The proof of Theorem 4 does contain a substantive gap that is worth flagging: Bergeron's separability lemma is applied with G = SU(2,1), while Theorem 3 shows that the cusp groups for families (3) and (4) must contain antiholomorphic isometries, and the final 'intersection with the first factor of Lambda' does not obviously contain the diagonal image of rho_3(pi_1(N)). However, a missing or incorrect argument is not a circular reduction: no equation is defined in terms of the target conclusion, and no fitted quantity is renamed as a prediction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

There are no fitted numerical parameters. The proofs rely almost entirely on imported structural theorems about arithmetic and non-arithmetic complex hyperbolic lattices, which is normal for this field. The paper's genuine contribution is the explicit representation formulas and the eigenvalue obstruction.

assumptions (8)
  • standard math Generalized Bieberbach theorems (Auslander, Lee-Raymond): discrete cocompact subgroups of Nil ⋊ C are conjugate in the affine group when abstractly isomorphic.
    Stated as Theorem 5 in Section 2.1, used in Lemma 1, Theorem 3, and Theorem 2 to fix the holonomy normal form.
  • domain assumption The classification of compact Nil 3-manifold fundamental groups into seven infinite families from [Dek].
    Quoted in Section 2.1; every theorem is stated relative to this list.
  • domain assumption Description of cusp subgroups of Picard modular groups Gamma_infty(d) in Proposition 1 from [FP], [FFP], [PW].
    Used to know the vertical and horizontal translation parts and compare with non-arithmetic cusp groups.
  • domain assumption Every non-cocompact arithmetic lattice in PU(2,1) is commensurable to exactly one PU(2,1,O_d) and is contained in PU(2,1,E_d).
    Cited from [St2] Section 3.1; is the algebraic input for the eigenvalue obstruction in Theorem 2.
  • domain assumption Known non-arithmetic complex hyperbolic lattices exist, have the cusp groups listed in Lemma 4 and Corollaries 1-2, and are integral in the sense used in Theorem 4.
    Imported from [DPP1], [DPP2], [DFP]; Theorem 4 depends on these cusp groups and integrality.
  • domain assumption Bergeron's lemma: for an algebraic subgroup H of a linear algebraic group G and a finitely generated Gamma < G, H ∩ Gamma is separable in Gamma.
    Proposition 2 in Theorem 4 Step (2); used to promote immersions to embeddings.
  • domain assumption Theorem 8 of [M2]: a torsion-free virtually unipotent subgroup of an arithmetic lattice is contained in a torsion-free finite-index subgroup.
    Used in Theorem 4 Step (3) to remove torsion from the non-arithmetic cover.
  • domain assumption Every automorphism of Nil is induced by an isometry of H^2_C, so affine conjugation preserves the holomorphic/antiholomorphic dichotomy.
    Used in Theorems 2 and 3 to transfer rigidity statements from Isom(Nil) to PU(2,1) versus Isom(H^2_C); cited from [M1] Section 2.4.

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Cite this review

Pith. "Pith review of Nil 3-manifolds and cusps of complex hyperbolic surfaces." pith.science (2026). https://pith.science/paper/XT3HBUWV

@misc{pith2026241115345,
  author       = {Pith},
  title        = {Pith review of: Nil 3-manifolds and cusps of complex hyperbolic surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XT3HBUWV}},
  note         = {Machine review of arXiv:2411.15345}
}
read the original abstract

McReynolds showed that every compact Nil 3-manifold occurs as the cusp cross-section of some arithmetic complex hyperbolic 2-manifold. We classify which commensurability classes of cusped, arithmetic, complex hyperbolic 2-manifolds admit cusps with cross-section homeomorphic to a given compact Nil 3-manifold. In particular, there are some Nil 3-manifolds which occur as cusps in every such commensurability class, and some which only occur in a single commensurability class. We also show that every compact Nil 3-manifold occurs as the cusp cross-section of some non-arithmetic complex hyperbolic 2-manifold.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

33 extracted references · 31 canonical work pages

  1. [1]

    Auslander; Bieberbach's theorems on space groups and discrete uniform subgroups of Lie groups

    L. Auslander; Bieberbach's theorems on space groups and discrete uniform subgroups of Lie groups . Ann. of Math. (2) 71 (1960), 579--590. 

    work page 1960

  2. [2]

    Bergeron; Premier nombre de Betti et spectre du laplacien de certaines variétés hyperboliques

    N. Bergeron; Premier nombre de Betti et spectre du laplacien de certaines variétés hyperboliques . Enseign. Math. (2) 46 (2000), no. 1-2, 109--137

    work page 2000

  3. [3]

    S. Chen, L. Greenberg; Hyperbolic spaces , in Contributions to Analysis. Academic Press, New York (1974), 49--87

    1974

  4. [4]

    Dekimpe; Almost-Bieberbach groups: affine and polynomial structures

    K. Dekimpe; Almost-Bieberbach groups: affine and polynomial structures . Lecture Notes in Mathematics 1639, Springer-Verlag, Berlin, 1996. 

    work page 1996

  5. [5]

    Deraux, On subgroups finite index in complex hyperbolic lattice triangle groups

    M. Deraux, On subgroups finite index in complex hyperbolic lattice triangle groups . Exp. Math. 33 (2024), no.3, 456--481. 

    work page 2024

  6. [6]

    Deraux, E

    M. Deraux, E. Falbel, J. Paupert; New constructions of fundamental polyhedra in complex hyperbolic space . Acta Math. 194 (2005), 155--201

    work page 2005

  7. [7]

    Deraux, J.R

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

    work page 2016

  8. [8]

    Deraux, J.R

    M. Deraux, J.R. Parker, J. Paupert; New non-arithmetic complex hyperbolic lattices II . Michigan Math. J. 70 (2021), 133--205. 

    work page 2021

Show all 33 references
  1. [9]

    Deraux, M

    M. Deraux, M. Stover; One-cusped complex hyperbolic 2-manifolds. Preprint (2024). arXiv: 2409.08028 

    2024

  2. [10]

    Falbel, J.R

    E. Falbel, J.R. Parker; The geometry of the Eisenstein-Picard modular group . Duke Math. J. 131 (2006) 249--289. 

    2006

  3. [11]

    Falbel, G

    E. Falbel, G. Francsics, J.R. Parker; The geometry of the Gauss-Picard modular group . Math. Ann. 349 (2011), no. 2, 459--508. 

    2011

  4. [12]

    F. T. Farrell, S. Zdravkovska; Do almost flat manifolds bound . Michigan J. Math. 30 (1983) 199--208. 

    1983

  5. [13]

    Goldman; Complex Hyperbolic Geometry

    W.M. Goldman; Complex Hyperbolic Geometry. Oxford Mathematical Monographs. Oxford University Press (1999)

    1999

  6. [14]

    Goldman; Geometric structures on manifolds

    W.M. Goldman; Geometric structures on manifolds. Graduate Studies in Mathematics 227 . American Mathematical Society, Providence, RI, 2022. 

    2022

  7. [15]

    Gromov; Almost flat manifolds

    M. Gromov; Almost flat manifolds . J. Differential Geom. 13 (1978), 231--241. 

    1978

  8. [16]

    Kamishima; Cusp cross-sections of hyperbolic orbifolds by Heisenberg nilmanifolds I

    Y. Kamishima; Cusp cross-sections of hyperbolic orbifolds by Heisenberg nilmanifolds I . Geom. Dedicata 122 (2006), 33--49. 

    2006

  9. [17]

    Kamishima; Nonexistence of cusp cross-section of one-cusped complete complex hyperbolic manifolds II

    Y. Kamishima; Nonexistence of cusp cross-section of one-cusped complete complex hyperbolic manifolds II . Int. Math. Forum 2 (2007), no. 25-28, 1251--1258. 

    2007

  10. [18]

    Kolpakov, B

    A. Kolpakov, B. Martelli; Hyperbolic four-manifolds with one cusp . Geom. Funct. Anal. 23 (2013), no.6, 1903--1933

    2013

  11. [19]

    K.B. Lee, F. Raymond; Rigidity of almost crystallographic groups . Combinatorial methods in topology and algebraic geometry, Contemp. Math. 44 , 73--78. American Mathematical Society, Providence, RI, 1985. 

    1985

  12. [20]

    D. D. Long, A. W. Reid; On the geometric boundaries of hyperbolic 4-manifolds . Geometry & Topology 4 (2000), 171--178. 

    2000

  13. [21]

    D. D. Long, A. W. Reid; All flat manifolds are cusps of hyperbolic orbifolds . Algebraic and Geometric Topology 2 (2002), 285--296. 

    2002

  14. [22]

    D. B. McReynolds; Peripheral separability and cusps of arithmetic hyperbolic orbifolds . Algebraic and Geometric Topology 4 (2004), 721--755. 

    2004

  15. [23]

    D. B. McReynolds; Controlling manifold covers of orbifolds. Math. Res. Lett. 16 (2009), no.4, 651--662. 

    2009

  16. [24]

    Paupert, P

    J. Paupert, P. Will; Real reflections, commutators and cross-ratios in complex hyperbolic space. Groups Geom. Dyn. 11 (2017), 311--352. 

    2017

  17. [25]

    Rohlin; A three-dimensional manifold is the boundary of a four-dimensional one

    V.A. Rohlin; A three-dimensional manifold is the boundary of a four-dimensional one . Doklady Akad. Nauk SSSR (N.S.) 81 (1951), 355--357

    1951

  18. [26]

    Scott; The geometries of 3-manifolds

    P. Scott; The geometries of 3-manifolds . Bull. London Math. Soc. 15 (1983), 401--487. 

    1983

  19. [27]

    Sell; Cusps and commensurability classes of hyperbolic 4-manifolds

    C. Sell; Cusps and commensurability classes of hyperbolic 4-manifolds . Algebraic and Geometric Topology 23 , No. 8 (2023), 3405--3434. 

    2023

  20. [28]

    Stover; On the number of ends of rank one locally symmetric spaces

    M. Stover; On the number of ends of rank one locally symmetric spaces . Geometry & Topology 17 (2013), 905--924. 

    2013

  21. [29]

    Stover; Volumes of Picard modular surfaces

    M. Stover; Volumes of Picard modular surfaces . Proc. Amer. Math. Soc. 39 (2011), no. 9, 3045--3056. 

    2011

  22. [30]

    Witte Morris; Introduction to arithmetic groups

    D. Witte Morris; Introduction to arithmetic groups. Deductive Press, 2015. Available at: https://arxiv.org/abs/math/0106063

    2015 arXiv

  23. [31]

    K. Dekimpe. Almost-Bieberbach groups: affine and polynomial structures. Lecture Notes in Mathematics 1639, Springer-Verlag, Berlin, 1996

    1996

  24. [32]

    D. B. McReynolds. Peripheral separability and cusps of arithmetic hyperbolic orbifolds. Algebraic and Geometric Topology Vol. 4, pp. 721-755, 2004

    2004

  25. [33]

    C. Sell. Cusps and commensurability classes of hyperbolic 4-manifolds. Preprint, 2020. To appear in Algebraic and Geometric Topology

    2020

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