Pith. sign in

REVIEW 2 major objections 4 minor 42 references

Geometric transition from hyperbolic to anti-de Sitter structures in dimension four

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Hyperbolic and anti-de Sitter structures are joined continuously for the first time in dimension four.

desk verdict First genuine hyperbolic-to-AdS geometric transition in dimension four, built from explicit polytopes; the main theorem is probably right, but the load-bearing vertex enumeration is not shipped with the paper. read the letter →

arxiv 1908.05112 v3 pith:MFO3CM5C submitted 2019-08-14 math.GT math.DG

Stefano Riolo , Andrea Seppi This is my paper · ORCID
classification math.GTmath.DG MSC 57M5053C5051M1052B10
keywords geometrictransitionanti-deSittergeometryhyperbolichalf-pipecone-manifoldsprojectivepolytopescuboctahedronfoamsingularities
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

This paper claims that geometric transition—a continuous path of geometric structures passing through an intermediate half-pipe geometry—happens in dimension four, not just three. For any hyperbolic 3-manifold $N$ that finitely orbifold-covers the ideal right-angled cuboctahedron, the 4-manifold $N\times S^1$ carries a $C^1$ family of simple projective cone-manifold structures, singular along a compact foam, that are hyperbolic with decreasing cone angles for positive time, half-pipe at time zero, and anti-de Sitter with increasing boost magnitude for negative time. All of them collapse to the complete hyperbolic structure of $N$ at the transition point. A sympathetic reader should care because this is the first evidence that higher-dimensional rigidity does not block Lorentzian and Riemannian geometries from being continuously joined by explicit polytope constructions.

What carries the argument

The load-bearing object is the deforming projective 4-polytope $P_t$, a family of 4-polytopes defined by 22 half-spaces with coefficients depending on $t$. For $t>0$, $P_t$ is a finite-volume hyperbolic polytope whose non-right dihedral angles tend to $\pi$ as $t\to 0$, collapsing to the ideal right-angled cuboctahedron $C\subset H^3$; the paper extends the same half-space pattern to $t<0$, where $P_t$ is an anti-de Sitter polytope with spacelike and timelike facets, and shows the rescaled family $r_{|t|}(P_t)$ converges to a half-pipe polytope at $t=0$. The proof that all of these have one constant combinatorics—46 vertices, with 12 on the boundary and 34 inside—is done by a computer-assisted enumeration (Lemma 7.8) rather than by a written calculation, and every gluing step depends on it. The gluing pairs facets of copies of $P_t$ following an orbifold covering of the cuboctahedron and doubles the result, producing $N\times S^1$ with the foam singularity, while explicitly computed limits of rescaled reflections supply the half-pipe orbifold structure.

What would settle it

Run the exhaustive vertex enumeration for $r_{|t|}(P_t)$ over a dense sample of $t\in(-1,0)$ and at $t=0$: solve the linear systems defining intersections of bounding hyperplanes, check whether each solution lies in the polytope, and compare the face-incidence poset with the positive-time case. Finding any vertex count different from 46, or any edge or face incidence that changes with $t$, would disprove Proposition 7.3 and with it the claimed constancy of the topology of $X_t$.

Watch

Extended reading notes

Core claim

The central result, Theorem 1.1, is a $C^1$ family $\{\sigma_t\}_{t\in(-\epsilon,\epsilon]}$ of simple projective cone-manifold structures on $X=N\times S^1$, singular along a compact foam $\Sigma$. At $t=\epsilon$ the structure is a complete finite-volume hyperbolic orbifold with cone angles $\pi$; for $t>0$ it is a hyperbolic cone structure with cone angles $\alpha_t\in[\pi,2\pi)$ decreasing as $t\to 0^+$; at $t=0$ it is a half-pipe structure with spacelike singularity; and for $t<0$ it is an anti-de Sitter structure with spacelike singularity of magnitude $\beta_t\in(-\infty,0)$ increasing as $t$ moves away from zero. The paper obtains these structures by taking a known one-parameter family of hyperbolic 4-polytopes that collapses to the ideal right-angled cuboctahedron, proving an analogous family of anti-de Sitter polytopes exists for negative times, rescaling to get a half-pipe limit, and gluing copies of the polytope according to an orbifold covering $N\to C$. The singular locus is a foam, a 2-complex locally modelled on the cone over the 1-skeleton of a tetrahedron.

Load-bearing premise

The load-bearing premise is that the rescaled polytope has exactly the same 46-vertex combinatorics for every $t$ in the interval, including the anti-de Sitter and half-pipe regimes; the proof is omitted and deferred to a computer worksheet, so an undiscovered change of combinatorics would break the gluing and the statement of Theorem 1.1.

Editorial extensions

If this is right

  • Geometric transition from hyperbolic to anti-de Sitter structures is not confined to dimension three; explicit four-dimensional examples exist, with a foam as singular locus.
  • The method produces finite-volume non-compact examples whose cusp sections themselves transition from Euclidean to Minkowski geometry through Galilean geometry.
  • The local models near the singular foam give the first four-dimensional links supporting a transition from spherical to HS cone structures, via an intermediate half-pipe cone structure.
  • The construction is flexible: the paper notes it extends to any cuboctahedral hyperbolic 3-manifold with chequerboard-preserving facet pairings, and to a double cover for every cuboctahedral manifold.

Reading between the lines

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

  • If the constant-combinatorics lemma is correct, the same half-space deformation pattern could probably be adapted to other right-angled ideal 3-polytopes that admit collapsing 4-dimensional deformations, producing transitions beyond the cuboctahedron.
  • The paper's explicit handling of the half-pipe reflection ambiguity suggests a general principle: a geometric transition is fixed only when one records the limits of the reflection holonomies, not just the projective limit of the polytopes.
  • A testable next step, which the paper does not settle, is whether the foam singularity is necessary or whether an embedded surface can support a four-dimensional transition; the paper notes no such surface-singularity deformations are currently known.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper constructs, in dimension four, explicit examples of geometric transition from hyperbolic to anti-de Sitter structures through half-pipe geometry. The main object is a family of projective 4-polytopes P_t, defined by 22 explicit half-spaces depending on a parameter t in (-1, 1/sqrt(3)]. For t>0 these are the Kerckhoff--Storm hyperbolic polytopes; the paper extends the family to t<0 as anti-de Sitter polytopes, applies the rescaling r_{|t|}, and shows that the rescaled polytopes converge at t=0 to a half-pipe polytope. Copies of P_t are then glued according to an orbifold cover of the ideal right-angled cuboctahedron, and the resulting space is doubled to obtain the 4-manifold X=N x S^1. The main theorem, Theorem 1.1, states that these glued structures form a C^1 family of simple projective cone-manifold structures with singular locus a compact foam, realizing hyperbolic cone structures for t>0, a half-pipe structure at t=0, and AdS cone structures for t<0, with the cone angles and boost magnitudes tending to the stated limiting values and the structures collapsing to a hyperbolic 3-manifold N.

Significance. If the construction is fully justified, this is the first geometric transition from hyperbolic to anti-de Sitter structures in dimension four, a significant extension of Danciger's three-dimensional work. The paper provides concrete, explicit half-space coordinates, derives the dihedral angle formulas, and proves the C^1 convergence of rescaled reflections in Lemma 7.15. The AdS and half-pipe sides are derived independently rather than obtained by fitting, and the gluing construction is detailed. The description of cusp geometry, including the Galilean transition on horospherical sections, is an additional strength. However, the central combinatorial-geometric assertions for the new cases t<=0 are not fully proved in the text: Lemma 7.9's printed proof is incomplete, and Lemma 7.8's vertex enumeration is deferred to an old arXiv appendix and an unspecified Sage worksheet. Since Proposition 7.3 and all subsequent gluing statements rely on these lemmas, the main theorem is conditional on repairing these gaps.

major comments (2)
  1. [§7.2, Lemma 7.9] The proof of Lemma 7.9 as printed analyzes only the inequalities (9) and (11), omitting the t-dependent inequalities (10). The inference 'By summing the first two equations and using the third, we get y1=y2=sqrt(2)/2' is not valid: summing the first two inequalities gives only y1+y2 <= sqrt(2), and the inequalities y2 <= y1 <= sqrt(2)/2 do not force equality. For example, at t=-0.9 the affine point (0.7, 0.6, 0.6, -0.509...) satisfies (9), (11), and the boundary equation y1^2+y2^2+y3^2-t^2 y4^2=1, but it violates the first inequality of (10). Thus the stated conclusion r_{|t|}(Q_t) cap boundary X^4_t = {[2:sqrt(2):sqrt(2):0:0]} is not established by the argument given. Since Lemma 7.9 is used in the proof of Proposition 7.3 to prove r_{|t|}(P_t) subset X^4_t cap A^4, the containment for t<0 and t=0 is not established as written. The statement may be true, but the proof must be repaired.
  2. [§7.2, Lemma 7.8] Lemma 7.8 asserts that r_{|t|}(P_t) has 46 vertices for all t in I, with 12 on the boundary of X^4_t and 34 in its interior. For t in I- union {0} the proof is replaced by the statement that the number of computations is 'terribly big', with details deferred to the old arXiv version [RSb, Appendix A] and an unspecified Sage worksheet. This is load-bearing: Proposition 7.3, Proposition 7.10, Proposition 7.13, and the gluing/homeomorphism arguments all use the constancy of the combinatorics and the 12/34 split. If the vertex count or the boundary/interior split changed with t, the topology of X_t or the singular foam could change and Theorem 1.1 would fail. A complete, machine-checkable certificate, or the actual code and its full output, must be included in the final version; a reference to an earlier arXiv version of the same paper is not a substitute.
minor comments (4)
  1. [Abstract] The word 'contribuitions' in the abstract is a typo and should be 'contributions'.
  2. [Section 7, opening paragraph] The phrase 'giving as a byproduct also rigours to the assertions' should read 'giving as a byproduct also rigor to the assertions'.
  3. [§7.2, after Proposition 7.3] The sentence 'In contrast with Pt, the polytope Pt is simple' appears to contain a typo; please clarify which polytope is meant, since both occurrences are printed identically.
  4. [§7.2, Lemma 7.8] If the Sage computation is retained, the final version should provide a stable identifier or repository URL for the worksheet, together with the exact version of Sage used, so that the verification is reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the AdS and half-pipe deformations are computed from explicit half-space data and independent geometric formulas, not fitted to the claimed transition.

full rationale

The paper's derivation chain is not circular. The hyperbolic side is imported from the published, external work of Kerckhoff–Storm and Martelli–Riolo, while the anti-de Sitter and half-pipe sides are constructed from the same explicit list of half-spaces in Table 2, with containment, combinatorics, dihedral angles, reflections, and holonomy limits computed directly from the quadratic forms q1, q−1, q0 via Lemmas 4.4, 4.6, 4.8, 4.10 and Lemma 7.15. No parameter is fitted to the target conclusion: the cone angles θt and magnitudes ϕt are derived, not assigned, and the C1 dependence follows from the explicit matrix entries of the rescaled reflections. Proposition 7.3, the load-bearing step, is supported by Lemma 7.8 and Lemma 7.9. Lemma 7.8's vertex enumeration is admittedly deferred to an old arXiv version of the same paper and an unspecified Sage worksheet, and Lemma 7.9's printed proof does omit the t-dependent inequalities (10) of the rescaled fundamental domain. These are gaps in verification or exposition, not circular reductions: the asserted combinatorics is computed from the half-space data rather than assumed, and no equation of the claimed transition is used as an input. The self-citations [RSb] and [RSa] point to computational details and a posteriori uniqueness, but the central existence claim does not reduce to those citations; the hyperbolic input is independently published, and the AdS/HP geometry is derived. Hence no circular step satisfying the quoted-evidence standard is present, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters appear in the paper; the parameter t is a deformation variable and all half-space coordinates are explicit. The only non-standard input is the computer-assisted vertex enumeration of Lemma 7.8, whose details are delegated to an old version of the paper. No new physical or mathematical entities are postulated beyond the existing frameworks of half-pipe geometry, foam singularities, and projective cone-manifolds.

assumptions (4)
  • standard math Kerckhoff-Storm [KS10] and Martelli-Riolo [MR18] establish the hyperbolic polytope P_t for t > 0: finite volume, constant combinatorics, facet and vertex description.
    The hyperbolic side of the deformation is imported from published theorems; the AdS extension and half-pipe limit are new work.
  • standard math Selberg's Lemma and Malcev's Theorem provide torsion-free finite-index subgroups of the reflection group of the cuboctahedron, giving orbifold covers N to C.
    Used in Section 7.6 to produce concrete 3-manifolds N that orbifold-cover the ideal right-angled cuboctahedron.
  • standard math Vinberg's theory of acute-angled hyperbolic polytopes is used in the hyperbolic case for the vertex enumeration of Lemma 7.8 and in [MR18].
    The hyperbolic case avoids heavy computation through Vinberg theory; the AdS and half-pipe cases require the omitted computer check.
  • ad hoc to paper The Sage vertex enumeration in Lemma 7.8 correctly and exhaustively lists the 46 vertices of r_{|t|}(P_t) for all t in the interval.
    This is the load-bearing computational step; the proof is omitted in the current version and no code is shipped, so the result is taken as an unverified external computation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Geometric transition from hyperbolic to anti-de Sitter structures in dimension four." pith.science (2026). https://pith.science/paper/MFO3CM5C

@misc{pith2026190805112,
  author       = {Pith},
  title        = {Pith review of: Geometric transition from hyperbolic to anti-de Sitter structures in dimension four},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFO3CM5C}},
  note         = {Machine review of arXiv:1908.05112}
}
read the original abstract

We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional ideal cuboctahedron. We show the existence of a similar family of collapsing anti-de Sitter polytopes, and join the two deformations by means of an opportune half-pipe orbifold structure. The desired examples of geometric transition are then obtained by gluing copies of the polytope.

Figures

Figures reproduced from arXiv: 1908.05112 by the authors.

Figure 1
Figure 1. A cuboctahedron in R3 . The ideal right-angled cuboctahedron in H3 can be seen as a cusped hyperbolic 3-orbifold [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. In red, the local models of a foam Σ, seen as open cones over some graphs (drawn in black). From left to right, a neighbourhood in Σ of a point in a 2-, 1-, and 0-stratum of Σ, respectively. Note that the third local model includes the other two. A foam in a 4-manifold is somehow the analogue of a trivalent graph in a 3-manifold. The cuboctahedron, drawn in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The transition from the ball model of Hn to the hyperboloid model of AdS n, in an affine chart (for n = 3). therefore conjugates the isometry group of Hn to the isometry group of X n t : Isom(X n t ) = rtIsom(H n )r −1 t . The following lemma says that half-pipe geometry is a “limit” of hyperbolic geometry: Lemma 2.9 ([CDW18, FS19]). When t → 0 +, the closure Xn t converges to HPn in the Haus￾dorff topology of S n, … view at source ↗
Figures from the paper (24 more)
Figure 4
Figure 4. Figure 4: Hyperbolic, half-pipe, and Anti-de Sitter horospheres in an affine chart. Definition 3.1. A horosphere in Hn is a smooth surface H ⊂ Hn which is orthogonal to all the geodesics with the same endpoint p ∈ ∂Hn. A horosphere in AdS n is a smooth timelike surface H ⊂ AdS n…
Figure 5
Figure 5. Figure 5: In an affine chart for Anti-de Sitter space, the two possibilities (above and below in the same figure) for the configuration of H and H0 as in Lemma 4.6 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: The two possibilities for the intersection of two timelike planes in AdS 3 : a timelike (left) or spacelike (right) line. where Hc is a half-space of S n−1 defined by the condition α0x0+. . .+αn−1xn−1 ≤ 0. Hence in the affine chart A n, H is the product of a half-space…
Figure 7
Figure 7. Figure 7: Hyperplanes in the affine (cylindric) model of HPn: on the left, two space￾like hyperplanes, on the right, a degenerate hyperplane. number, we can assume H = (α0 : . . . : αn−1 : ±1) . For instance, if αn = 1, we get that H is defined by the condition α0x0 + . . . αn−1…
Figure 8
Figure 8. Figure 8: A sheaf of hyperplanes in HPn (on the left) corresponds to the points of R1,n−1 lying on a spacelike line ` (on the right). Viceversa, the intersection of the sheaf is in bijection with the spacelike hyperplanes containing `. Recall that we denote by qb the quadratic f…
Figure 9
Figure 9. Figure 9: The argument of Proposition 4.15: Minkowski reflections in hyperplanes parallel to w⊥ (in grey) leave set-wise invariant every spacelike hyperplane (like P1 and P2 in the figure) having normal vector in w⊥ ∩ Hn−1 (this intersection is pictured as a hyperbola here). The…
Figure 10
Figure 10. Figure 10: The link of a point in a hyperbolic, half-pipe, or Anti-de Sitter n-manifold. In the hyperbolic case (left), we have the round sphere S n−1 . In the AdS case (right), the link is called HSn−1 , and is subdivided into two timelike regions (copies of Hn−1 ), one spaceli…
Figure 11
Figure 11. Figure 11: The minimally twisted 6-chain link in the 3-sphere. Its complement M is hyperbolic, and can be tessellated by four ideal right-angled octahedra. By 0-surgery on the two red components, we get a Dehn filling X of M homeomorphic to S0,4 × S 1 , where S0,4 is a 4-times p…
Figure 12
Figure 12. Figure 12: The polyhedron Oθ ⊂ H3 . The white dots represent ideal vertices, the black edges are right-angled, and the red edges have dihedral angle θ ∈ (0, π). As θ → 0, the red edges disappear, and we have the original ideal octahedron O0. As θ → π, the polyhedron collapses to…
Figure 13
Figure 13. Figure 13: A movie of the collapse of the polyhedron Oθ in an affine chart (Klein model of H3 ), from the ideal octahedron O0 to the ideal quadrilateral Q.  −|t| : − √ 2|t| : 0 : −1  ,  −1 : − √ 2 : 0 : +t  ,  −|t| : 0 : − √ 2|t| : +1 ,  −1 : 0 : − √ 2 : −t  ,  −|t| : +…
Figure 14
Figure 14. Figure 14: In an affine chart, the rescaled path of polyhedra for t ≤ 0. In the left figure (t=0), the polyhedron is in half-pipe space and the triangular faces are degenerate (vertical). In the middle picture, an AdS polyhedron with timelike triangular faces and spacelike quadr…
Figure 15
Figure 15. Figure 15: A facet Fi− , i− ∈ {0−, . . . , 7−}, of the 4-polytope of Section 7 (see the end of Section 7.2 for the notation). The white dots represent ideal vertices, the black edges are right-angled, and the red edges have dihedral angle θ ∈ [ π 2 , π). As θ → π, the polyhedron…
Figure 16
Figure 16. Figure 16: The collapse of the polyhedron Fϑ of Section 6.2, in the Klein model of H3 . The leftmost polyhedron turns out to be also the rescaled limit, inside HP3 . As s → 0 + (resp. s → 0 −), we have ϑs → 2π (resp. φs → 0) and the hyperbolic (resp. AdS) structures on X r Σ deg…
Figure 17
Figure 17. Figure 17: A facet Fi+ , i+ ∈ {0+, . . . , 7+}, of the 4-polytope Pt. The white dots represent ideal vertices, the black edges are right-angled, and the yellow edges have some other varying dihedral angle. The three red pentagons are ridges of Pt of type Ri+j+ , and have varying…
Figure 18
Figure 18. Figure 18: The links of the vertices of Pt (see Proposition 7.13). When t ∈ I + (resp. t ∈ I−) the link of an ideal vertex is a Euclidean (resp. Minkowski) right paralleleped, and the link of a finite vertex is a spherical tetrahedron (resp. de Sitter tetrahedron with spacelike …
Figure 19
Figure 19. Figure 19: The link of an ideal vertex of Pt, obtained by intersecting Pt with a horosphere. The further intersection with H3 , which is constant in t, is a rectangle (shaded in the picture). When t → 0, the rectangular parallepiped collapses to this rectangle [PITH_FULL_IMAGE:…
Figure 20
Figure 20. Figure 20: After rescaling, the geometry of the link of an ideal vertex transitions from Euclidean (left) to Minkoskian (right), via Galilean geometry (centre). The intersection with the fixed copy of H3 is shaded. This is an example of the transition explained in Section 3.2. 7…
Figure 21
Figure 21. Figure 21: The ideal right-angled cuboctahedron C = Pt ∩ H3 = P0. A quadrilateral face with label X ∈ {A, . . . , F } coincides with FX ∩ H3 , while a triangular face with label i ∈ {0, . . . , 7} coincides with the ridge Ri+i− of Pt. representation depends C 1 on the parameter …
Figure 22
Figure 22. Figure 22: The union C0 = F0+ ∪ F2+ ∪ F4+ ∪ F6+ (resp. C1 = F1+ ∪ F3+ ∪ F5+ ∪ F7+ ) is an ideal right-angled cuboctahedron, pleated along the 6 red pentagons Ri+j+ (each with 3 blue edges and 2 yellow edges in the picture). The facets of C0 are divided as follows: 4 ideal triang…
Figure 23
Figure 23. Figure 23: The effect of the gluing on the link `v of a vertex v of the link LV (which is depicted in [PITH_FULL_IMAGE:figures/full_fig_p045_23.png]
Figure 24
Figure 24. Figure 24: The effect of the gluing on the link LV of a finite vertex V of Pt (see the proof of Proposition 7.28). At each line we see the effect on a different class of vertices (see [PITH_FULL_IMAGE:figures/full_fig_p046_24.png]
Figure 25
Figure 25. Figure 25: The link of a point in a 2-stratum of Σt ⊂ Xt is a cone 3-sphere with singular locus an unknotted circle (drawn in red). The geometry is spherical when t ∈ I + (left), and HS when t ∈ I− (right). In the HS case, the two balls (one internal and one external) are copies…
Figure 26
Figure 26. Figure 26: The link of a point in an edge (i.e. a 1-stratum) of Σt ⊂ Xt, t 6= 0, is a cone 3-sphere with singular locus an unknotted theta-graph [PITH_FULL_IMAGE:figures/full_fig_p047_26.png]
Figure 27
Figure 27. Figure 27: The link of a vertex (i.e. a 0-stratum) of Σt ⊂ Xt, t 6= 0, is a cone 3-sphere with singular locus an unknotted complete graph on four vertices [PITH_FULL_IMAGE:figures/full_fig_p047_27.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 40 canonical work pages

  1. [1]

    E. M. Andreev. Convex polyhedra in L oba c evski spaces. Mat. Sb. (N.S.) , 81 (123):445--478, 1970

    work page 1970

  2. [2]

    E. M. Andreev. Convex polyhedra of finite volume in L oba c evski space. Mat. Sb. (N.S.) , 83 (125):256--260, 1970

    work page 1970

  3. [3]

    Transitional geometry

    Norbert A'Campo and Athanase Papadopoulos. Transitional geometry. Sophus Lie and Felix Klein: the Erlangen program and its impact in mathematics and physics , 23:217, 2015

    work page 2015

  4. [4]

    Collisions of particles in locally A d S spacetimes I

    Thierry Barbot, Francesco Bonsante, and Jean-Marc Schlenker. Collisions of particles in locally A d S spacetimes I . L ocal description and global examples. Comm. Math. Phys. , 308(1):147--200, 2011

    work page 2011

  5. [5]

    Quasi- F uchsian co- M inkowski manifolds, 2018

    Thierry Barbot and Fran c ois Fillastre. Quasi- F uchsian co- M inkowski manifolds, 2018. To appear in book ``In the tradition of Thurston'', arXiv:1801.10429 https://arxiv.org/abs/1801.10429

    arXiv 2018

  6. [6]

    Bridson and A

    M.R. Bridson and A. H \"a fliger. Metric Spaces of Non-Positive Curvature . Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2011

    work page 2011

  7. [7]

    Geometrization of 3-dimensional orbifolds

    Michel Boileau, Bernhard Leeb, and Joan Porti. Geometrization of 3-dimensional orbifolds. Ann. of Math. (2) , 162(1):195--290, 2005

    work page 2005

  8. [8]

    Limits of geometries

    Daryl Cooper, Jeffrey Danciger, and Anna Wienhard. Limits of geometries. Trans. Amer. Math. Soc. , 370:6585--6627, 2018

    work page 2018

Show all 42 references
  1. [9]

    Hodgson, and Steven P

    Daryl Cooper, Craig D. Hodgson, and Steven P. Kerckhoff. Three-dimensional orbifolds and cone-manifolds , volume 5 of MSJ Memoirs . Mathematical Society of Japan, Tokyo, 2000. With a postface by Sadayoshi Kojima

    2000

  2. [10]

    Geometric structures on orbifolds and holonomy representations

    Suhyoung Choi. Geometric structures on orbifolds and holonomy representations. Geom. Dedicata , 104:161--199, 2004

    2004

  3. [11]

    Choi, G.-S

    S. Choi, G.-S. Lee, and L. Marquis. Convex projective generalized D ehn filling. Ann. Sci. \'Ec. Norm. Sup\'er. , 53:217--266, 2020

    2020

  4. [12]

    Geometric transition: from hyperbolic to AdS geometry

    Jeffrey Danciger. Geometric transition: from hyperbolic to AdS geometry . PhD thesis, Stanford University, 2011

    2011

  5. [13]

    A geometric transition from hyperbolic to anti-de S itter geometry

    Jeffrey Danciger. A geometric transition from hyperbolic to anti-de S itter geometry. Geom. Topol. , 17(5):3077--3134, 2013

    2013

  6. [14]

    Ideal triangulations and geometric transitions

    Jeffrey Danciger. Ideal triangulations and geometric transitions. J. Topol. , 7(4):1118--1154, 2014

    2014

  7. [15]

    Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions

    Fran c ois Fillastre and Andrea Seppi. Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions. In Eighteen essays in non- E uclidean geometry , volume 29 of IRMA Lect. Math. Theor. Phys. , pages 321--409. Eur. Math. Soc., Z\" u rich, 2019

    2019

  8. [16]

    Degeneration and regeneration of hyperbolic structures on three-manifolds (foliations, D ehn surgery)

    Craig David Hodgson. Degeneration and regeneration of hyperbolic structures on three-manifolds (foliations, D ehn surgery) . ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)--Princeton University

    1986

  9. [17]

    Regenerating singular hyperbolic structures from S ol

    Michael Heusener, Joan Porti, and Eva Su\' a rez. Regenerating singular hyperbolic structures from S ol. J. Differential Geom. , 59(3):439--478, 2001

    2001

  10. [18]

    Hyperbolic four-manifolds with one cusp

    Alexander Kolpakov and Bruno Martelli. Hyperbolic four-manifolds with one cusp. Geom. Funct. Anal. , 23(6):1903--1933, 2013

    1903

  11. [19]

    Singular hyperbolic structures on pseudo-Anosov mapping tori

    Kenji Kozai. Singular hyperbolic structures on pseudo-Anosov mapping tori . PhD thesis, Stanford University, 2013

    2013

  12. [20]

    Hyperbolic structures from sol on pseudo-anosov mapping tori

    Kenji Kozai. Hyperbolic structures from sol on pseudo-anosov mapping tori. Geometry & Topology , 20(1):437--468, 2016

    2016

  13. [21]

    Kerckhoff and Peter A

    Steven P. Kerckhoff and Peter A. Storm. From the hyperbolic 24-cell to the cuboctahedron. Geom. Topol. , 14(3):1383--1477, 2010

    2010

  14. [22]

    Geometric conemanifold structures on T _ p/q , the result of p/q surgery in the left-handed trefoil knot T

    Mar \' a Teresa Lozano and Jos \'e Mar \' a Montesinos-Amilibia. Geometric conemanifold structures on T _ p/q , the result of p/q surgery in the left-handed trefoil knot T . Journal of Knot Theory and Its Ramifications , 24(12):1550057, 2015

    2015

  15. [23]

    On the degeneration of some 3-manifold geometries via unit groups of quaternion algebras

    Mar \' a Teresa Lozano and Jos \'e Mar \' a Montesinos-Amilibia. On the degeneration of some 3-manifold geometries via unit groups of quaternion algebras. Revista de la Real Academia de Ciencias Exactas, F \' sicas y Naturales. Serie A. Matem \'a ticas , 109(2):669--715, 2015

    2015

  16. [24]

    A small closed convex projective 4-manifold via D ehn filling

    Gye-Seon Lee, Ludovic Marquis, and Stefano Riolo. A small closed convex projective 4-manifold via D ehn filling. 2019. arXiv:1910.11649 https://arxiv.org/abs/1801.10429

    2019 arXiv

  17. [25]

    Coxeter group in H ilbert geometry

    Ludovic Marquis. Coxeter group in H ilbert geometry. Groups Geom. Dyn. , 11(3):819--877, 2017

    2017

  18. [26]

    McMullen

    Curtis T. McMullen. The G auss- B onnet theorem for cone manifolds and volumes of moduli spaces. Amer. J. Math. , 139(1):261--291, 2017

    2017

  19. [27]

    Hyperbolic D ehn filling in dimension four

    Bruno Martelli and Stefano Riolo. Hyperbolic D ehn filling in dimension four. Geom. Topol. , 22(3):1647--1716, 2018

    2018

  20. [28]

    Regenerating hyperbolic and spherical cone structures from E uclidean ones

    Joan Porti. Regenerating hyperbolic and spherical cone structures from E uclidean ones. Topology , 37(2):365--392, 1998

    1998

  21. [29]

    Regenerating hyperbolic cone structures from N il

    Joan Porti. Regenerating hyperbolic cone structures from N il. Geom. Topol. , 6:815--852, 2002

    2002

  22. [30]

    Regenerating hyperbolic cone 3-manifolds from dimension 2

    Joan Porti. Regenerating hyperbolic cone 3-manifolds from dimension 2. Ann. Inst. Fourier (Grenoble) , 63(5):1971--2015, 2013

    1971

  23. [31]

    Deforming E uclidean cone 3-manifolds

    Joan Porti and Hartmut Weiss. Deforming E uclidean cone 3-manifolds. Geom. Topol. , 11:1507--1538, 2007

    2007

  24. [32]

    Character varieties of a transitioning C oxeter 4-orbifold

    Stefano Riolo and Andrea Seppi. Character varieties of a transitioning C oxeter 4-orbifold. In preparation

  25. [33]

    Old ar X iv version of the present paper

    Stefano Riolo and Andrea Seppi. Old ar X iv version of the present paper. arXiv:1908.05112v1 https://arxiv.org/abs/1908.05112v1

    1908 arXiv

  26. [34]

    New hyperbolic 4-manifolds of low volume

    Stefano Riolo and Leone Slavich. New hyperbolic 4-manifolds of low volume. Alg. Geom. Topol. , 19(5):2653--2676, 2019

    2019

  27. [35]

    Examples of geometric transition from dimension two to four

    Andrea Seppi. Examples of geometric transition from dimension two to four. To appear, A ctes du s \'e minaire T h \'e orie S pectrale et G \'e om \'e trie, 26 pages , 2020

    2020

  28. [36]

    Limits of quasi- F uchsian groups with small bending

    Caroline Series. Limits of quasi- F uchsian groups with small bending. Duke Math. J. , 128(2):285--329, 2005

    2005

  29. [37]

    S ageMath, the S age M athematics S oftware S ystem ( V ersion 8.1) , 2017

    The Sage Developers . S ageMath, the S age M athematics S oftware S ystem ( V ersion 8.1) , 2017. www.sagemath.org https://www.sagemath.org

    2017

  30. [38]

    W. P. Thurston. The geometry and topology of three-manifolds. Electronic version 1.1, link http://library.msri.org/nonmsri/gt3m, 1979

    1979

  31. [39]

    Thurston

    William P. Thurston. Shapes of polyhedra and triangulations of the sphere. In The E pstein birthday schrift , volume 1 of Geom. Topol. Monogr. , pages 511--549. Geom. Topol. Publ., Coventry, 1998

    1998

  32. [40]

    Steve J. Trettel. Families of geometries, real algebras, and transitions . PhD thesis, University of C alifornia, S anta B arbara, 2019

    2019

  33. [41]

    Hyperbolic reflection groups

    E B Vinberg. Hyperbolic reflection groups. Russian Mathematical Surveys , 40(1):31, 1985

    1985

  34. [42]

    I. M. Yaglom. A simple non- E uclidean geometry and its physical basis . Springer-Verlag, New York-Heidelberg, 1979. An elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, Translated from the Russian by Abe Shenitzer, W...

    1979

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.