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arxiv: 2605.03161 · v1 · submitted 2026-05-04 · 🧮 math.GT · math.DG· math.GR

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Parabolic-preserving deformations of cusped hyperbolic lattices

Julien Paupert, Pierre Will, Samuel A. Ballas

Pith reviewed 2026-05-08 02:39 UTC · model grok-4.3

classification 🧮 math.GT math.DGmath.GR
keywords parabolic-preserving deformationscusped hyperbolic latticesfigure-eight knot groupBianchi groupsbending deformationsSU(n,1)Zariski-dense representationshyperbolic n-manifolds
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The pith

Cusped hyperbolic n-manifolds for n at least 3 admit one-parameter families of parabolic-preserving deformations into SU(n,1).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates deformations of non-cocompact lattices from SO(n,1) into SU(n,1) and SO(n+1,1). It defines strongly parabolic-preserving representations as those where parabolic subgroups remain parabolic and discrete. The figure-eight knot group is shown to admit a one-parameter family of Zariski-dense parabolic-preserving deformations into SU(3,1), with further deformations into SU(2,2). Infinitely many Bianchi groups have strongly parabolic-preserving bending deformations into SU(3,1), though none do into SO(4,1). The central general result is that for every n at least 3, infinitely many non-commensurable cusped hyperbolic n-manifolds have one-parameter families of parabolic-preserving deformations into SU(n,1).

Core claim

The figure-eight knot group admits a one-parameter family of Zariski-dense parabolic-preserving deformations into SU(3,1), with further deformations into SU(2,2). Infinitely many Bianchi groups admit strongly parabolic-preserving bending deformations into SU(3,1), while none are strongly parabolic-preserving in SO(4,1). For any n greater than or equal to 3, there exist infinitely many non-commensurable cusped hyperbolic n-manifolds whose hyperbolic representation admits a 1-parameter family of parabolic-preserving deformations into SU(n,1).

What carries the argument

Strongly parabolic-preserving representations, in which the images of parabolic subgroups remain parabolic and discrete.

Load-bearing premise

Deformations remain discrete and faithful only if the parabolic subgroups stay parabolic and discrete.

What would settle it

Explicit computation of the image of a parabolic generator under a generic deformed representation showing that it fails to remain parabolic or discrete for the figure-eight knot group or a Bianchi group.

read the original abstract

We study deformations of non-cocompact lattices of ${\rm SO}(n,1)$ into ${\rm SU}(n,1)$ and ${\rm SO}(n+1,1)$. A necessary condition for these deformations to remain discrete and faithful (when $n \geqslant 3$) is for the parabolic subgroups to remain parabolic and discrete; we call such representations \emph{strongly parabolic-preserving}. We show that the figure-eight knot group admits a one-parameter family of Zariski-dense parabolic-preserving deformations into ${\rm SU}(3,1)$, with further deformations into ${\rm SU}(2,2)$. We also study the \emph{bending deformations} of the Bianchi groups (seen as subgroups of ${\rm SO}(3,1)$) along the modular surface into ${\rm SU}(3,1)$ and ${\rm SO}(4,1)$, and show that infinitely many of them are strongly parabolic-preserving in ${\rm SU}(3,1)$, while none are strongly parabolic-preserving in ${\rm SO}(4,1)$. Finally, for any $n \geqslant 3$, we show that there exist infinitely many non-commensurable cusped hyperbolic $n$-manifolds whose corresponding hyperbolic representation admits a 1-parameter family of parabolic-preserving deformations into ${\rm SU}(n,1)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies deformations of non-cocompact lattices in SO(n,1) into SU(n,1) and SO(n+1,1). It proves existence of a one-parameter family of Zariski-dense parabolic-preserving deformations of the figure-eight knot group into SU(3,1), with further deformations into SU(2,2). It analyzes bending deformations of Bianchi groups, establishing that infinitely many are strongly parabolic-preserving in SU(3,1) while none are in SO(4,1). For every n ≥ 3 it constructs infinitely many non-commensurable cusped hyperbolic n-manifolds whose hyperbolic representations admit 1-parameter families of parabolic-preserving deformations into SU(n,1). The notions of parabolic-preserving and strongly parabolic-preserving representations are defined explicitly, with the latter noted as necessary (but not claimed sufficient) for discreteness and faithfulness when n ≥ 3.

Significance. If the proofs hold, the work supplies both concrete low-dimensional examples and a uniform existence result for parabolic-preserving deformations across all dimensions n ≥ 3. The explicit treatment of the figure-eight knot group and the bending construction for Bianchi groups, together with the separation of the weaker parabolic-preserving condition from the stronger discrete-parabolic condition, provides useful test cases for questions about flexibility versus rigidity in higher-rank representation varieties. The uniform construction for arbitrary n is a notable structural contribution to the study of cusped hyperbolic manifolds and their deformations.

minor comments (3)
  1. The abstract states that the figure-eight family is Zariski-dense, but the general n-manifold construction does not mention Zariski-density; if this property is not claimed in general, a brief clarifying sentence would prevent misreading.
  2. In the bending-deformation analysis for Bianchi groups, the proof that infinitely many are strongly parabolic-preserving in SU(3,1) relies on a count of modular surfaces or cusp parameters; an explicit reference to the arithmetic condition used for the count would strengthen readability.
  3. Notation for the target groups (SU(n,1) versus SO(n+1,1)) is consistent, but the manuscript would benefit from a single table or diagram summarizing which deformations are shown to be strongly parabolic-preserving versus merely parabolic-preserving.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript, their positive assessment of its significance, and their recommendation for minor revision. No specific major comments or criticisms were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines parabolic-preserving and strongly parabolic-preserving representations explicitly as a necessary condition for potential discreteness and faithfulness, then constructs explicit one-parameter families and bending deformations for the figure-eight knot group, Bianchi groups, and general cusped hyperbolic n-manifolds that satisfy these conditions via geometric arguments. No step reduces a claimed existence result or deformation family to a fitted parameter, self-referential definition, or load-bearing self-citation by the paper's own equations. The constructions are presented as independent of the target claims, with Zariski-density shown separately, making the derivation self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background from Lie group theory and hyperbolic geometry; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of parabolic subgroups in the Lie groups SO(n,1) and SU(n,1) and their action on hyperbolic space.
    Invoked to define the notion of strongly parabolic-preserving representations.

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Works this paper leans on

34 extracted references · 1 canonical work pages

  1. [1]

    Ballas; Deformations of noncompact projective manifolds

    S.A. Ballas; Deformations of noncompact projective manifolds . Algebr. Geom. Topol. 14 (2014), no. 5, 2595--2625

    2014

  2. [2]

    Ballas; Finite volume properly convex deformations of the figure-eight knot

    S.A. Ballas; Finite volume properly convex deformations of the figure-eight knot . Geom. Dedicata 178 (2015), 49--73

    2015

  3. [3]

    Ballas, J

    S.A. Ballas, J. Danciger, G.-S. Lee; Convex projective structures on non-hyperbolic three-manifolds . Geom. Topol. 22 (2018), no. 3, 1593--1646

    2018

  4. [4]

    Ballas, D.D

    S.A. Ballas, D.D. Long; Constructing thin subgroups commensurable with the figure-eight knot group . Algebr. Geom. Topol. 15 (2015), no. 5, 3011--3024

    2015

  5. [5]

    Ballas, L

    S.A. Ballas, L. Marquis; Properly convex bending of hyperbolic manifolds . Groups Geom. Dyn. 14 (2020), no. 2, 653--688

    2020

  6. [6]

    Bart and K

    A. Bart and K. Scannell; The generalized cuspidal cohomology problem . Canad. J. Math. 58 (2006), no. 4, 673--690

    2006

  7. [7]

    Bergeron, F, Haglund, D

    N. Bergeron, F, Haglund, D. Wise; Hyperplane sections in arithmetic hyperbolic manifolds . J. Lond. Math. Soc. (2) 83 (2011), no. 2, 431--448

    2011

  8. [8]

    Bers; Simultaneous uniformization , Bull

    L. Bers; Simultaneous uniformization , Bull. Amer. Math. Soc. (66). (1960), 94--97

    1960

  9. [9]

    Bowditch; Spaces of geometrically finite representations

    B. Bowditch; Spaces of geometrically finite representations . Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 2, 389--414

    1998

  10. [10]

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

    1974

  11. [11]

    Cooper, D.D

    D. Cooper, D.D. Long and M.B. Thistlethwaite; Flexing closed hyperbolic manifolds . Geom. Topol. 11 (2007), 2413--2440

    2007

  12. [12]

    Cooper, D.D

    D. Cooper, D.D. Long and M.B. Thistlethwaite; Computing varieties of representations of hyperbolic 3-manifolds into SL(4, R ) . . J. of Experimental Math. vol. 15 (2006) pp. 291 -- 305

    2006

  13. [13]

    Dunbar, G.R

    W.D. Dunbar, G.R. Meyerhoff; Volumes of hyperbolic 3-orbifolds . Indiana Univ. Math. J. 43 (1994), no. 2, 611--637

    1994

  14. [14]

    Fine; Algebraic theory of the Bianchi groups

    B. Fine; Algebraic theory of the Bianchi groups. Monogr. Textbooks Pure Appl. Math., 129 . Marcel Dekker, Inc., New York, 1989, viii+249 pp

    1989

  15. [15]

    Feng, K, Stange; Notes on the spin homomorphism

    Y. Feng, K, Stange; Notes on the spin homomorphism . Expositional notes for Apollonian Circle Packings, June 2014, Mittag-Leffler Institute. Available at: https://math.colorado.edu/\ kstange/papers/notes-Spin.pdf

    2014

  16. [16]

    Garland, M.S

    H. Garland, M.S. Raghunathan; Fundamental domains for lattices in ( -)rank 1 semisimple Lie groups , Ann. of Math. (2) 92 (1970), 279--326

    1970

  17. [17]

    Goldman; Complex Hyperbolic Geometry

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

    1999

  18. [18]

    Gongopadhyay, S

    K. Gongopadhyay, S. Lawton; Invariants of pairs in SL (4, ) and SU (3,1) . Proc. Amer. Math. Soc. 145 (2017), no. 11, 4703--4715

    2017

  19. [19]

    Guichard; Groupes plong\'es quasi-isom\'etriquement dans un goupe de Lie

    O. Guichard; Groupes plong\'es quasi-isom\'etriquement dans un goupe de Lie . Math. Ann. 330 (2004), no. 2, 331--351

    2004

  20. [20]

    Guilloux, T

    A. Guilloux, T. Weisman; Limits of limit sets in rank-one symmetric spaces . Preprint (2024). Available at: https://arxiv.org/abs/2407.04301

    work page arXiv 2024

  21. [21]

    Hatcher and W

    A. Hatcher and W. Thurston; Incompressible surfaces in 2-bridge knot complements . Invent. Math. 79 (1985), no. 2, 225--246

    1985

  22. [22]

    Heusener and J

    M. Heusener and J. Porti; Infinitesimal projective rigidity under Dehn filling . Geom. Topol. 15 (2011), no. 4, 2017--2071

    2011

  23. [23]

    Johnson and J

    D. Johnson and J. Millson; Deformation Spaces Associated to Compact Hyperbolic Manifolds , in Papers in Honor of G.D. Mostow on His Sixtieth Birthday (Roger Howe, ed.). Birkh\"auser, Progress in Mathematics 67 (1986), 48--106

    1986

  24. [24]

    Kapovich; Deformations of representations of discrete subgroups of SO (3,1)

    M. Kapovich; Deformations of representations of discrete subgroups of SO (3,1) . Math. Annalen. 299 (1994), 341--354

    1994

  25. [25]

    McReynolds

    D.B. McReynolds. A.W. Reid, and M. Stover; Collisions at infinity in hyperbolic manifolds , Math. Proc. Cambridge Philos. Soc. 155 (2013), no. 3, 459--463

    2013

  26. [26]

    Parker, I.D

    J.R. Parker, I.D. Platis; Complex hyperbolic quasi-Fuchsian groups , in: Geometry of Riemann Surfaces. London Mathematical Society Lecture Notes 368 (2010) 309--355

    2010

  27. [27]

    Paupert, M

    J. Paupert, M. Thistlethwaite; Complex hyperbolic and projective deformations of small Bianchi groups . Geom. Dedicata 217 , 98 (2023)

    2023

  28. [28]

    Scannell; Local rigidity of hyperbolic 3-manifolds after Dehn surgery

    K. Scannell; Local rigidity of hyperbolic 3-manifolds after Dehn surgery . Duke Math. J. 114 (2002), no. 1, 1--14

    2002

  29. [29]

    Schwartz; Complex hyperbolic triangle groups , from: ÒProceedings of the International Congress of Mathematicians, Vol

    R.E. Schwartz; Complex hyperbolic triangle groups , from: ÒProceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002)Ó, Higher Ed. Press, Beijing (2002) 339--349

    2002

  30. [30]

    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. Geom. Topol. 17 (2013), no. 2, 905--924

    2013

  31. [31]

    Swan; Generators and relations for certain special linear groups

    R.G. Swan; Generators and relations for certain special linear groups . Adv. Math. 6 (1971), 1--77

    1971

  32. [32]

    Teranishi; The ring of invariants of matrices

    Y. Teranishi; The ring of invariants of matrices . Nagoya Math. J. 104 (1986), 149--161

    1986

  33. [33]

    Thurston; The Geometry and Topology of Three-Manifolds

    W.P. Thurston; The Geometry and Topology of Three-Manifolds. Electronic version 1.1 (2002). Available at: http://www.msri.org/publications/books/gt3m/

    2002

  34. [34]

    Weil; Discrete subgroups of Lie groups (II) , Ann

    A. Weil; Discrete subgroups of Lie groups (II) , Ann. of Math. (2) 75 (1962), 578--602

    1962