Lorentz spacetimes of constant curvature

Given f : π1S → SO(2,1)0 and M = L+/f(π1S) as in Proposition 2, and any bounded neighbourhood U of 0 in H1(π1S,R2+1), there exists C such that, letting M_C = { (x,y,t) : -t^2 + x^2 + y^2 ≤ -C }/f(π1S), there exists a family of flat Lorentz metrics on M_C parametrized by U such that the spacetime M_C(u) corresponding to u ∈ U has holonomy ρ : π1M_C → ISO(2,1) such that qρ = f and ρ(γ)x = f(γ)x + tγ

“Proposition 3. Given f : π 1 S → SO (2 , 1) 0 and M = L + /f ( π 1 S ) as in proposition 2, and any bounded neighbourhood U of 0 in H 1 ( π 1 S, R 2+1 ) there exists C such that, letting M C = { ( x, y, t ) : -t 2 + x 2 + y 2 ≤ -C } /f ( π 1 S ) , there exists a family of flat Lorentz metrics on M C parametrized by U”

• cites [14] Matsumoto S. (1987). Some remarks on foliated S 1 bundles. Invent. Math. 90(2): 343–358 sourceNot checked — no full text of the source
• cites [15] Milnor J. (1977). On fundamental groups of complete affinely flat manifolds. Adv. Math. 25(2): 178–187 sourcePartly supported

  Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured
• cites [16] Thurston, W.P.: The Geometry and Topology of Three-manifolds. Lecture notes, Princeton University (1978)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [17] Lok, L.: Deformations of locally homogeneous spaces and Kleinian groups. Thesis, Columbia University (1984)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Labourie [52] has shown that the spacetimes in Proposition 16 are foliated by spacelike convex surfaces of constant curvature.

“Labourie [52] has shown that the spacetimes in proposition 16 are foliated by spacelike convex surfaces of constant curvature; these surfaces are dual to surfaces of constant curvature in hyperbolic space.”

• cites [52] Goldman W.M. (1985). Nonstandard Lorentz space forms. J. Differential Geom. 21(2): 301–308 sourceCould not verify

  In the section “Nonstandard Lorentz Space Forms” (page unavailable), Goldman discusses deformations of complete Lorentz structures, not foliations by spacelike convex constant-curvature surfaces. Searches for Labourie, Proposition 16, and such surfaces found no passage addressing the attributed result; the paper’s discussion of a foliation concerns a flat bundle instead. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

The map from the space of projective structures to the space of holonomy representations is not a covering map.

“and that given one projective structure there are infinitely many others with the same holonomy, but the map from the space of projective structures to the space of holonomy representations is not a covering map [47, 48].”

• cites [47] Carrière Y., Dal’bo F. (1989). Généralisations du premier théorème de Bieberbach sur les groupes cristallographiques. Enseign. Math. (2) 35(3–4): 245–262Not checked yet

  Not checked in the free run.

• cites [48] Tits J. (1972). Free subgroups in linear groups. J. Algebra 20: 250–270 sourceCould not verify

  The passages in §3.6 concern projective transformations and their attracting and repulsing subspaces; those in §4 concern linear groups. A search for projective structures, holonomy representations, and the covering-map claim found no passage addressing that map. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

After passing to a finite cover of the surface using [11], we may assume that the surface has a decomposition along curves with purely hyperbolic holonomy into submanifolds S_i whose holonomy groups are quasifuchsian.

“Then after passing to a finite cover of the surface using [11], we may assume that the surface has a decomposition along curves with purely hyperbolic holonomy into submanifolds S i whose holonomy groups are quasifuchsian.”

• cites [11] Benzécri J.-P. (1960). Sur les variétés localement affines et localement projectives. Bull. Soc. Math. France 88: 229–332 sourceCould not verify

  Our copy discusses compact locally projective surfaces in “Les résultats antérieurs” and decompositions of convex sets elsewhere, but neither passage addresses the claimed surface decomposition. Searches for finite covers, curves with purely hyperbolic holonomy, and quasifuchsian holonomy found no supporting passage; page numbers were not available. (LLM, conf=0.72)

  Experimental — accuracy not yet measured

There is another family of de Sitter spacetimes which have a Riemann surface at past infinity.

“There is another family of spacetimes which have a Riemann surface at past infinity.”

Proposition 5: Suppose M is a flat Lorentz manifold containing a closed spacelike future directed surface S of genus g > 1. There exists a standard spacetime M' containing a future directed strictly convex surface S' of genus g and such that S and S' have the same holonomy. Furthermore ̃S' can be chosen to lie in the future of ̃S.

“Proposition 5. Suppose M is a flat Lorentz manifold containing a closed spacelike future directed surface S of genus g > 1 . There exists a standard spacetime M ′ containing a future directed strictly convex surface S ′ of genus g and such that S and S ′ have the same holonomy. Furthermore ˜ S ′ can be chosen to lie in the future of ˜ S .”

The Euler class of qρ : π1S → SO(2,1)0 is the Euler class of the tangent bundle, that is, 2 - 2g if S is oriented so the positive normal to S is future pointing.

“So the Euler class of the homomorphism qρ : π 1 S → SO (2 , 1) 0 is the Euler class of the tangent bundle, that is, 2 -2 g if S is oriented so the positive normal to S is future pointing.”

Lemma 1: Suppose i : F → R2+1 is a spacelike immersion of a connected surface F and F is complete in the induced metric. Then F is the graph of some function h : R2 → R, and in particular is embedded.

“Lemma 1. Suppose i : F → R 2+1 is a spacelike immersion of a connected surface F and F is complete in the induced metric. Then F is the graph of some function h : R 2 → R , and in particular is embedded.”

Corollary 1: If a spacetime M contains a closed spacelike surface S of negative Euler characteristic then S has an isochronous neighbourhood in M.

“Corollary 1. If a spacetime M contains a closed spacelike surface S of negative Euler characteristic then S has an isochronous neighbourhood in M .”

M in Proposition 2 can be chosen either future complete or past complete.

“Note that M can be chosen either future complete or past complete.”

Since ρ(π1S) acts properly discontinuously on R, and t is a proper function on R which is bounded below, R/ρ(π1S) contains no closed timelike curves.

“Since ρ ( π 1 S ) acts properly discontinuously on R , and t is a proper function on R which is bounded below, R/ρ ( π 1 S ) contains no closed timelike curves.”

A closed spacelike hypersurface S cannot have g = 0.

“A closed spacelike hypersurface S cannot have g = 0.”

In section 6 the paper constructs a family of de Sitter spacetimes, each of which is foliated by closed spacelike hypersurfaces.

“In section 6 we construct a family of de Sitter spacetimes, each of which is foliated by closed spacelike hypersurfaces, which conjecturally is the solution to the problem.”

Proposition 4: Let M be a flat Lorentz spacetime. Suppose M contains a closed spacelike surface S of genus g > 1. Then i) development embeds ̃S in R2+1 and ii) the coordinate function t is a proper function on ̃S.

“Proposition 4. Let M be a flat Lorentz spacetime. Suppose M contains a closed spacelike surface S of genus g > 1 . Then i ) development embeds ˜ S in R 2+1 and ii ) the coordinate function t is a proper function on ˜ S .”

The paper generalizes Carrière's theorem to compact Lorentzian manifolds with spacelike boundary.

“In section 8, we generalize this to the case of compact Lorentzian manifolds with spacelike boundary.”

Each de Sitter spacetime in that family corresponds to a projective structure on a Riemann surface at future infinity.

“Each spacetime corresponds to a projective structure on a Riemann surface at future infinity.”

If g > 1 then qρ : π1S → SO(2,1)0 is injective and qρ(π1S) is a discrete cocompact subgroup of SO(2,1)0.

“Proposition 1. If g > 1 then qρ : π 1 S → SO (2 , 1) 0 is injective and qρ ( π 1 S ) is a discrete cocompact subgroup of SO (2 , 1) 0 .”

• cites [6] Witten E. (1989). Topology-changing amplitudes in (2+1)-dimensional gravity. Nuclear Phys. B 323(1): 113–140 sourceNot checked — no full text of the source

This result generalizes to Lorentzian manifolds of constant negative curvature.

“This result generalizes to Lorentzian manifolds of constant negative curvature.”

By Lemma 1, S_u = dev( ̃T,u) is a convex surface.

“By Lemma 1, S u = dev ( ˜ T, u ) is a convex surface.”

Given a discrete and faithful f : π1S → SO(2,1)0, there is a flat Lorentz spacetime M with M = S × (0,∞) and holonomy ρ : π1M → ISO(2,1) equal to f.

“Proposition 2. Given a discrete and faithful f : π 1 S → SO (2 , 1) 0 , there is a flat Lorentz spacetime M with M = S × (0 , ∞ ) and holonomy ρ : π 1 M → ISO (2 , 1) equal to f .”

Proposition 6: If M is a spacetime with a closed spacelike surface S of negative Euler characteristic such that π1S = π1M, then there is a spacetime M'' containing a neighbourhood of S in M and containing a standard spacetime M'.

“Proposition 6. If M is a spacetime with a closed spacelike surface S of negative Euler characteristic such that π 1 S = π 1 M , then there is a spacetime M ′′ containing a neighbourhood of S in M and containing a standard spacetime M ′ .”

More generally, an R2-bundle with a flat GL(2,R) connection has Euler number at most g - 1 in absolute value.

“More generally, an R 2 -bundle with a flat GL (2 , R ) connection has Euler number at most g -1 in absolute value.”

Proposition 12: Given a closed oriented hyperbolic surface S and a measured geodesic lamination on S, there is a corresponding flat Lorentz manifold M and an embedding of S in M such that M is a domain of dependence of S and the linear holonomy of S is the map H : π1S → SO(2,1)0 which is the holonomy of S considered as a hyperbolic surface.

“Proposition 12. Given a closed oriented hyperbolic surface S and a measured geodesic lamination on S , there is a corresponding flat Lorentz manifold M and an embedding of S in M such that M is a domain of dependence of S and the linear holonomy of S is the map H : π 1 S → SO (2 , 1) 0 which is the holonomy of S considered as a hyperbolic surface.”

Proposition 11b: Any timelike linear function on R2+1 which increases into the future or the past according as ̃S is future or past directed is bounded below on D.

“b) Any timelike linear function on R 2+1 which increases into the future or the past according as ˜ S is future or past directed is bounded below on D .”

If L contains isolated leaves, the transverse measure may be smoothed out over a neighbourhood of each isolated leaf and the resulting function f' is a homeomorphism.

“If L contains isolated leaves, the transverse measure may be smoothed out over a neighbourhood of each isolated leaf and the resulting function say f ′ is a homeomorphism”

Equality in the preceding null-or-spacelike statement holds only if f(p)=p.

“Equality holds only if f ( p ) = p .”

Every null ray in X parallel to l1 has basepoint in rq.

“Moreover every null ray in X parallel to l 1 has basepoint in rq”

If f(p)=f(q) and q∈A00, then p=q.

“This establishes that if f ( p ) = f ( q ) and q ∈ A 00 then p = q .”

There is no open subset of X where the direction of the null rays is nonconstant and varies continuously.

“There is no open subset of X where the direction of the null rays is nonconstant and varies continuously.”

The map a∘t^{-1}:H^2→X maps almost all tangent planes to the countable set C.

“It follows that the map a ◦ t -1 : H 2 → X maps almost all tangent planes to the countable set C .”

• AssumptionRests on an assumption: A geodesic lamination on a hyperbolic surface has measure 0.

Each de Sitter manifold in the family of Proposition 16 admits a conformal compactification at future infinity.

“Each admits a conformal compactification at future infinity, and the projection π to P ( F ) is given by assigning to a given spacetime the projective structure on the Riemann surface at future infinity.”

Euler characteristic considerations imply that this finite set is empty.

“Euler characteristic considerations imply that this finite set is empty.”

̃S' is a translate of ̃S by a vector valued function discontinuous only across ̃L0, the sublamination of ̃L consisting of leaves of nonzero mass.

“˜ S ′ is a translate of ˜ S by a vector valued function which is discontinuous only across ˜ L 0 , the sublamination of ˜ L consisting of leaves of nonzero mass.”

Each component of T0 develops into a spacelike curve.

“Each component of T 0 develops into a spacelike curve.”

An infinite cyclic cover T' of T lies in the simply connected de Sitter space, which is the complement in the three-sphere of two antipodal round discs.

“Now an infinite cyclic cover T ′ of T lies in the simply connected de Sitter space, which is the complement in the three-sphere of two antipodal round discs.”

If the holonomy is purely translational, then since dev ̃T is spacelike and projects onto R2, the holonomy must map the two generators of Z ⊕ Z onto independent spacelike vectors.

“Now suppose the holonomy is purely translational. Since dev ˜ T is spacelike and projects onto R 2 , the holonomy must map the two generators of Z ⊕ Z onto independent spacelike vectors.”

Given a point (x,y) in Teich(S) × Teich(S) there exists a compact spacetime X(x,y) with boundary, homeomorphic to S × [0,1] by a homeomorphism taking any S × {x} to a closed spacelike hypersurface, which is locally modelled on anti de Sitter space, and which has left and right holonomies ρ_L, ρ_R determined by the point (x,y).

“Proposition 20. Given a point ( x, y ) in Teich ( S ) × Teich ( S ) there exists a compact spacetime X ( x, y ) with boundary, homeomorphic to S × [0 , 1] by a homeomorphism taking any S ×{ x } to a closed spacelike hypersurface, which is locally modelled on anti de Sitter space, and which has left and right holonomies ρ L , ρ R determined by the point ( x, y ) .”

There is a natural homotopy equivalence between M and M0.

“There is a natural homotopy equivalence between M and M 0 .”

Proposition 11e: Null rays contained in X and containing p correspond one-to-one to null supporting planes at p.

“e) Null rays contained in X and containing p correspond one-to-one to null supporting planes at p .”

After passing to a finite cover of the spacetime, we may assume that the holonomy is conjugate into the identity component of the diagonal subgroup of SL2R.

“So after passing to a finite cover of the spacetime we may assume that the holonomy is conjugate into the identity component of the diagonal subgroup of SL 2 R .”

Lemma 3: The support of the transverse measure on L is all of L.

“Lemma 3. The support of the transverse measure on L is all of L .”

The quotient spacetime M is future complete.

“π 1 S acts and the quotient spacetime M evidently is future complete”

For any positive t, the surface {t}×F̃ is the dual in de Sitter space of the surface in H3 at distance t from the locally convex bent surface x(F̃) in the direction toward infinity.

“For any positive t , the surface { t } × ˜ F is the dual in de Sitter space of the surface in H 3 at distance t from the locally convex bent surface x ( ˜ F ) in the direction toward infinity.”

Let M be a de Sitter manifold which is a small neighbourhood of a closed spacelike torus. Then (perhaps after a change of time orientation) there is a torus with affine structure such that M embeds in the spacetime associated to the torus by Proposition 16.

“Proposition 18. Let M be a de Sitter manifold which is a small neighbourhood of a closed spacelike torus. Then (perhaps after a change of time orientation) there is a torus with affine structure such that M embeds in the spacetime associated to the torus by proposition 16.”

̃M is contained in the domain of dependence of ̃S'.

“So ˜ M is contained in the domain of dependence of ˜ S ′ .”

Given any complete spacelike surface A ⊂ X, there is a spacelike plane in X whose closure in X ∪ Q ⊂ RP3 is disjoint from the closure of A.

“Sublemma 1. Given any complete spacelike surface A ⊂ X , there is a spacelike plane in X whose closure in X ∪ Q ⊂ RP 3 is disjoint from the closure of A .”

After an arbitrarily small change in the holonomy of each of the boundary circles, the eigenvalue re^{iθ} of the holonomy at a fixed point on CP1 will have θ a rational multiple of π.

“After an arbitrarily small change in the holonomy of each of the boundary circles, the eigenvalue re iθ of the holonomy at a fixed point on CP 1 will have θ a rational multiple of π .”

There is a second family of past complete spacetimes which differs only by the covering transformation of the covering of the non simply connected de Sitter space by simply connected de Sitter space.

“(There is a second family of past complete spacetimes which differs only by the covering transformation of the covering of the non simply connected de Sitter space by simply connected de Sitter space.)”

q is in the domain of dependence of f( ̃S)= ̃S''.

“So q is in the domain of dependence of f ( ˜ S ) = ˜ S ′′ .”

By another small change in the geometric structure, one of the summands of Z⊕Z is in the kernel of the holonomy.

“Then by another small change in the geometric structure we have (in an appropriate basis) one of the summands of Z ⊕ Z is in the kernel of the holonomy.”

If a finite cover of M embeds in a domain of dependence of an affine structure, then M itself does.

“Also, if a finite cover of M embeds in a domain of dependence of an affine structure, M itself does.”

M is a domain of dependence.

“So M is a domain of dependence.”

For each hyperbolic structure on S, there is a bijection from the space ML(S) of measured geodesic laminations on S to H^1(π1S,R^{2+1}).

“We now have, for each hyperbolic structure on S , a bijection from the space ML ( S ) of measured geodesic laminations on S to H 1 ( π 1 S, R 2+1 ).”

Proposition 7: In cases 1) and 2) the linear holonomy is trivial, and a neighbourhood of T embeds in a complete spacetime.

“Proposition 7. In cases 1) and 2) the linear holonomy is trivial, and a neighbourhood of T embeds in a complete spacetime.”

Proposition 13: Given a spacetime M which is the domain of dependence of a closed spacelike surface S of genus g>1, the geometry of the frontier of the universal cover determines a measured geodesic lamination on S in the hyperbolic structure determined by its linear holonomy, and this correspondence is inverse to that of Proposition 12.

“Proposition 13. Given a spacetime M which is the domain of dependence of a closed spacelike surface S of genus g > 1 the geometry of the frontier of the universal cover determines a measured geodesic lamination on S in the hyperbolic structure on S determined by its linear holonomy, and this correspondence is inverse to that of Proposition 12.”

The holonomy of M in Proposition 18 cannot be conjugate into the subgroup SU(2) of SL2C.

“First of all, the holonomy of M cannot be conjugate into the subgroup SU (2) of SL 2 C .”

The surface {t}×F̃ is the dual of a locally strictly convex surface which has t as its smaller principal curvature at each point.

“So it is the dual of a locally strictly convex surface which has t as its smaller principal curvature at each point.”

There is a plane (necessarily spacelike) disjoint from graph(h) ⊂ Q.

“Lemma 5. There is a plane (necessarily spacelike) disjoint from graph ( h ) ⊂ Q .”

The projection π to P(F) in Proposition 16 is given by assigning to a given spacetime the projective structure on the Riemann surface at future infinity.

“Each admits a conformal compactification at future infinity, and the projection π to P ( F ) is given by assigning to a given spacetime the projective structure on the Riemann surface at future infinity.”

Proposition 11c: At any boundary point p, all supporting planes are null or spacelike.

“c) At any boundary point p , all supporting planes are null or spacelike.”

M is a flat Lorentz manifold with a closed spacelike convex surface made from pieces of S and product annuli.

“Then M is a flat Lorentz manifold with a closed spacelike convex surface (made from pieces of S and product annuli).”

There is an isometric S^1 action on de Sitter space which commutes with the holonomy of T and fixes an axis, the great circle which passes through the two fixed points of the action of the holonomy of T on either of the spheres at infinity of de Sitter space.

“There is an isometric S 1 action on de Sitter space which commutes with the holonomy of T and fixes an axis, that is, the great circle which passes through the two fixed points of the action of the holonomy of T on either of the spheres at infinity of de Sitter space.”

E is dense in the complement of the ruled regions of X.

“So taking q = P ( m ) ∩ X , we have seen that E is dense in the complement of the ruled regions of X .”

Given a spacelike disc D which meets no timelike or null geodesic in more than one point, let F(D), respectively P(D), be the set of x ∈ X such that every past pointing, respectively future pointing null ray or timelike geodesic through x meets D. Then P(D) ∩ F(D) = D.

“Lemma 7. a) Given a spacelike disc D which meets no timelike or null geodesic in more than one point, let F ( D ) , respectively P ( D ) , be the set of x ∈ X such that every past pointing, respectively future pointing null ray or timelike geodesic through x meets D . Then P ( D ) ∩ F ( D ) = D .”

The element t of H^1(π1S,R^{2+1}) represented by M equals sum_i a_i t_i, where t_i is the extension by zero of PD(C_i)⊗e_i.

“Then t = ∑ i a i · t i where t i is the extension by zero of the cohomology class (with support in an open annular neighbourhood N ( C i )) PD ( C i ) ⊗ e i where PD is Poincar´ e duality and e i is the unit spacelike vector fixed by the holonomy of C i .”

Every point in D is in some set F(r).

“So every point in D is in some set F ( r ).”

These spacetimes are maximal domains of dependence essentially by construction.

“These spacetimes are maximal domains of dependence essentially by construction.”

If r≠s, then F(r) and F(s) are disjoint unless r and s lie on a single null ray.

“Claim. Suppose r = s . Then F ( r ) and F ( s ) are disjoint unless r and s lie on a single null ray.”

Future directed domains of dependence form a family parametrized by the bundle of cohomology spaces V=H^1(π1S,R^{2+1})→Teich(S).

“We have shown that (future directed) domains of dependence form a family parametrized by the bundle of cohomology spaces V = H 1 ( π 1 S, R 2+1 ) → Teich ( S )”

Proposition 8: In case 3) a neighbourhood of T in M embeds in a standard spacetime (possibly after a change of time orientation).

“Proposition 8. In case 3) a neighbourhood of T in M embeds in a standard spacetime (possibly after a change of time orientation).”

Proposition 15: Given a spacetime M=D( ̃S), there does not exist a spacetime M' strictly containing M such that the development of ̃M' does not meet the ruled part of X.

“Proposition 15. Given a spacetime M = D ( ˜ S ) , there does not exist a spacetime M ′ strictly containing M such that the development of ˜ M ′ does not meet the ruled part of X .”

If the holonomy of M were conjugate into SU(2), then projection onto CP1 along the family of lines meeting at the fixed point inside H3 of the subgroup conjugate to SU(2) would define a Riemannian structure on T of constant positive curvature, which is impossible for a closed surface of positive genus.

“If it was, then projection onto CP 1 along the family of lines which meet at the fixed point inside H 3 of the subgroup conjugate to SU (2) would define a Riemannian structure on T of constant positive curvature, which is impossible for a closed surface of positive genus.”

The linear holonomy qρ : Z ⊕ Z → SO(2,1)0 has abelian image.

“The linear holonomy qρ : Z ⊕ Z → SO (2 , 1) 0 has abelian image.”

If P and Q are nondegenerate Lorentzian planes through the origin in R^{2+1} whose intersection is timelike or null, then they divide the hyperbolic plane into three regions E,F,G with the stated closure properties.

“Suppose P and Q are nondegenerate Lorentzian planes through the origin in R 2+1 . Suppose the line of intersection of P and Q is timelike or null. Then P and Q divide the hyperbolic plane (the hyperboloid of unit timelike future pointing vectors) into three regions E,F,G where the closure of E meets P but not Q , the closure of F meets P and Q , and the closure of G meets Q but not P .”

The union of the parallel regions is a closed subset of X.

“The union of the parallel regions is a closed subset of X , because it is the intersection with X of the closure in D ∪ X of the preimage by t of the union of the isolated leaves of L , and t is continuous.”

Each point y on T has a unique tangent plane, because the support plane at a(y) is parallel to the tangent plane at y.

“Each point y on T has a unique tangent plane, because the support plane at a ( y ) is parallel to the tangent plane at y .”

The cocycle constructed in Proposition 12 depends continuously on the measured geodesic lamination.

“It is clear that the cocycle constructed in proposition 12 depends continuously on the measured geodesic lamination”

The holonomy ρ = (ρ_L, ρ_R): π1S → PSL2R × PSL2R satisfies the condition that ρ_L, ρ_R have Euler class 2 - 2g and therefore are isomorphisms to Fuchsian groups.

“Proposition 19. The holonomy ρ = ( ρ L , ρ R ) : π 1 S → PSL 2 R × PSL 2 R satisfies the condition that ρ L , ρ R have Euler class 2 -2 g and therefore are isomorphisms to Fuchsian groups.”

<A> is the fundamental group of an isolated leaf of the geodesic lamination determined by X.

“〈 A 〉 is the fundamental group of an isolated leaf of the geodesic lamination determined by X .”

If m,n are normals to P,Q oriented so m points into E and n points into F, then the scalar product <m,n> is positive.

“If m , n are normals to P, Q respectively, oriented so m points into E and n points into F , then the scalar product 〈 m , n 〉 is positive.”

Lemma 4: If L is a measured geodesic lamination on a closed surface for which L is the full support of the transverse measure, and if m1,m2 are distinct geodesics asymptotic to each other, then m1 and m2 are both boundary leaves of a complementary region Ai, and so each has transverse measure 0.

“Lemma 4. If L is a measured geodesic lamination on a closed surface for which L is the full support of the transverse measure, if m 1 , m 2 are distinct geodesics which are asymptotic to each other, then m 1 and m 2 are both boundary leaves of a complementary region A i , and so each of m 1 , m 2 has transverse measure 0 .”

f(D)' contains D'.

“So f ( D ) ′ ⊇ D ′ .”

In case (c) of Proposition 17, the universal cover of D(M) is the complement in RP2 of the union of the closure cl H2 and the tangent line to a point P ∈ S1 = ∂ cl H2.

“Then the universal cover ˜ D ( M ) of D ( M ) is the complement in RP 2 of the union of the closure cl H 2 and the tangent line to a point P ∈ S 1 = ∂ cl H 2 .”

Proposition 14: If X contains two parallel null rays l1 and l2, then X contains two null rays l0 and l3 parallel to l1 and l2 with basepoints r and q such that the segment rq is spacelike and lies in X and from each point in rq there is a unique null ray in X parallel to l1.

“Proposition 14. Suppose X contains two parallel null rays l 1 and l 2 . Then X contains two null rays l 0 and l 3 parallel to l 1 and l 2 with basepoints r and q such that the seqment rq is spacelike and lies in X and from each point in rq there is a unique null ray in X parallel to l 1 .”

Proposition 9: π1S acts properly discontinuously on D( ̃S).

“Proposition 9. π 1 S acts properly discontinuously on D ( ˜ S ) .”

For any genus g, there is a family of manifolds π: X → P(F) such that each fiber π^{-1}{a} is a de Sitter manifold which is a future complete maximal domain of dependence, foliated by strictly locally convex spacelike hypersurfaces.

“Proposition 16. For any genus g , there is a family of manifolds, π : X → P ( F ) , such that each fiber π -1 { a } is a de Sitter manifold which is a future complete maximal domain of dependence, foliated by strictly locally convex spacelike hypersurfaces.”

T is convex because the future of T is the union of the convex regions U(p)={y∈D:<y-p,y-p>>-1} where p∈X.

“T is convex because the future of T is the union of the convex regions U ( p ) = { y ∈ D : 〈 y -p, y -p 〉 &gt; -1 } where p ∈ X .”

Proposition 11a: The boundary X of a domain D = D( ̃S) of dependence in R2+1 is convex.

“Proposition 11. a) The boundary X of a domain D = D ( ˜ S ) of dependence in R 2+1 is convex.”

The space ML(S) of measured geodesic laminations is homeomorphic to R^{6g-6}.

“It follows that the space ML ( S ) of measured geodesic laminations is homeomorphic to R 6 g -6 .”

If L has closed leaves, product regions must be added to f(L+).

“If L has closed leaves, product regions must be added to f ( L + ).”

The interior of the union of the cones over the closures of the surfaces Aij+xh gives a spacetime ̃M unless L contains closed leaves.

“The interior of the union of the cones over the closures of the surfaces A ij + x h (where h is a path from A 00 to A ij ) gives a spacetime ˜ M unless L contains closed leaves.”

• AssumptionRests on an assumption: S - L is a union of finitely many regions A_i.

M̂ is foliated by the preimages by dev of the planes P_s.

“So ̂ M is foliated by the preimages by dev of the planes P s .”

In the proof of Proposition 27, M is complete.

“We conclude that M is complete.”

M can be thickened to a manifold with locally strictly convex boundary.

“Also, using proposition 21 M can be thickened to a manifold with locally strictly convex boundary.”

If M has a boundary component ∂iM then π1∂iM has index one in π1M.

“We conclude that if M has a boundary component ∂ i M then π 1 ∂ i M has index one in π 1 M .”

The proof of proposition 26 extends to show that E_x meets T_x P in a convex set.

“The proof of proposition 26 extends to show that E x meets T x P in a convex set.”

The boundary components of M must have negative Euler characteristic.

“The boundary components of M must have negative Euler characteristic by proposition 21.”

In the case of flat and anti de Sitter spacetimes containing a closed spacelike surface, a neighbourhood of the universal cover of the surface embeds in the model space.

“In the case of flat and anti de Sitter spacetimes containing a closed spacelike surface, we showed that a neighbourhood of the universal cover of the surface embeds in the model space.”

Choosing a subgroup R = SO(2) ⊂ G_L defines a time function t: X̃ → R such that the action of R permutes the planes P_s = t^{-1}(s).

“Choosing a subgroup R = ˜ SO (2) ⊂ G L defines a time function t : ˜ X → R such that the action of R permutes the planes P s = t -1 ( s ) , s ∈ R .”

Since M̂ has locally convex boundary, the subset Exp(E_x ∩ T_x P) ∩ U is convex.

“Since ̂ M has locally convex boundary, the subset Exp ( E x ∩ T x P ) ∩ U is convex.”

If M is a compact flat 2+1-dimensional orthochronous Lorentzian manifold with spacelike boundary and ∂M is nonempty, then M is a product ∂0M × [0,1] with each slice ∂0M × {t} spacelike.

“Otherwise M is a product ∂ 0 M × [0 , 1] with each slice ∂ 0 M ×{ t } spacelike.”

The inclusion of either boundary component of M induces an isomorphism of fundamental groups.

“Therefore the inclusion of either boundary component of M induces an isomorphism of fundamental groups.”

If M is a compact flat 2+1-dimensional orthochronous Lorentzian manifold with spacelike boundary and ∂M is empty, then M is complete.

“Proposition 26. Suppose M is a compact flat 2 + 1 -dimensional orthochronous Lorentzian manifold with spacelike boundary. Then if ∂M is empty M is complete.”

Given a point x ∈ M̂, the exponential is a diffeomorphism from the intersection of the domain E_x of the exponential with any spacelike plane in T_x M̂ to a convex set in a spacelike plane in M̂.

“Proposition 27. a) Given a point x ∈ ̂ M , the exponential is a diffeomorphism from the intersection of the domain E x of the exponential with any spacelike plane in the tangent space T x ̂ M to a convex set in a spacelike plane in ̂ M .”

The Weil-Petersson metric determines a second order operator on Teichmüller space, which measures the uncertainty of the translational part of the holonomy.

“The Weil-Petersson metric determines a second order operator on Teichm¨ uller space, which measures the uncertainty of the translational part of the holonomy.”

If M is a compact oriented orthochronous anti de Sitter manifold with spacelike boundary, then either M embeds in a domain of dependence or else M is a closed anti de Sitter manifold whose universal cover is a domain in X with locally convex frontier.

“b) If M is a compact oriented orthochronous anti de Sitter manifold with spacelike boundary, then either M embeds in a domain of dependence or else M is a closed anti de Sitter manifold whose universal cover is a domain in X with locally convex frontier.”

The preimage of a spacelike plane P is connected.

“Now we will show that the preimage of a spacelike plane P is connected.”

These anti de Sitter spacetimes are Lorentzian analogues of the hyperbolic manifolds corresponding to quasifuchsian groups.

“These spacetimes are Lorentzian analogues of the hyperbolic manifolds corresponding to quasifuchsian groups.”

The spacetimes in the anti de Sitter family have both an initial and a final singularity.

“The spacetimes in this family have both an initial and a final singularity.”

There is a natural family of maximal domains of dependence in anti de Sitter spacetimes, parametrized by the product of two copies of Teichmüller space, such that given any closed spacelike hypersurface in an anti de Sitter spacetime, a neighbourhood of the hypersurface embeds as an open subset of a unique member of the family.

“There is a natural family of maximal domains of dependence, parametrized by the product of two copies of Teichm¨ uller space, such that given any closed spacelike hypersurface in an anti de Sitter spacetime, a neighbourhood of the hypersurface embeds as an open subset of a unique member of the natural family of spacetimes.”

If ρ is discrete and faithful, then |e(ρ)| = 2g - 2.

“If ρ is discrete and faithful this is the tangent bundle (up to change of orientation) so | e | = 2 g -2.”

Conversely, if |e(ρ)| = 2g - 2 then ρ is discrete and faithful.

“Conversely Goldman shows in [6] that if | e ( ρ ) | = 2 g -2 then ρ is discrete and faithful”

• cites [6] Witten E. (1989). Topology-changing amplitudes in (2+1)-dimensional gravity. Nuclear Phys. B 323(1): 113–140 sourceNot checked — no full text of the source

In flat spacetimes, for a given genus, all domains of dependence form two families, each parametrized by the cotangent bundle of Teichmüller space.

“In the case of flat spacetimes, we show in section 2, 3, 4, 5 that for a given genus, all domains of dependence form two families, each parametrized by the cotangent bundle of Teichm¨ uller space.”

If y lies in the interior of the convex hull of ∂A, then no point in the dual plane of y can be in H(A).

“Suppose y lies in the interior of the convex hull of ∂A . Then no point in the dual plane of y can be in H ( A ).”

There is a plane disjoint from ∂A∞.

“There is a plane disjoint from ∂A ∞ by (the argument of) lemma 5.”

The boundary is F(φ) ∪ P(φ).

“So the boundary is F ( φ ) ∪ P ( φ ).”

After choosing a different product structure on SO(2)×SO(2) the intersections A_r become graphs of functions all of which have Lipschitz constant 1.

“After choosing a different product structure on SO (2) × SO (2) the intersections A r become graphs of functions all of which have Lipschitz constant 1.”

Z is the quotient of X(φ).

“It follows that Z is the quotient of X ( φ ).”

Given an orthochronous anti de Sitter spacetime which contains a product neighbourhood N(F) of a closed spacelike surface F, there is a smaller neighbourhood N'(F) which embeds in int(H(F(φ)))/π1F where φ conjugates the left and right holonomies of π1F.

“Proposition 21. Given an orthochronous anti de Sitter spacetime which contains a product neighbourhood N ( F ) of a closed spacelike surface F , then there is a smaller neighbourhood N ′ ( F ) which embeds in int ( H ( F ( φ )) ) /π 1 F where φ conjugates the left and right holonomies of π 1 F .”

The closure of the universal cover of ∂Z' is the union of graph(φ) and two locally convex spacelike surfaces.

“Then lemma 7 applies, and we deduce that the closure of the universal cover of ∂Z ′ is the union of graph ( φ ) and two locally convex spacelike surfaces.”

Gallo [42] shows that given a homomorphism from the fundamental group of a surface to SL2C with image not conjugate into SU(2) nor into the stabilizer of a point or pair of points in CP1, there is a decomposition of the surface into pairs of pants, such that the holonomy of any pair of pants is quasifuchsian.

“Gallo [42] actually shows that given a homomorphism from the fundamental group of a surface to SL 2 C with image not conjugate into SU (2) nor into the stabilizer of a point or pair of points in CP 1 , there is a decomposition of the surface into pairs of pants, such that the holonomy of any pair of pants is quasifuchsian.”

• cites [42] Drumm T.A., Goldman W.M. (1990). Complete flat Lorentz 3-manifolds with free fundamental group. Internat. J. Math. 1(2): 149–161 sourceNot checked — no full text of the source

Suppose E:H2∪RP1→H2∪RP1 is a left earthquake along a lamination λ with transverse measure μ. Then there is an isometry I from the hyperbolic plane to the future component of the convex hull of the graph of the boundary value homeomorphism φ of E.

“Proposition 23. Suppose E : H 2 ∪ RP 1 → H 2 ∪ RP 1 is a left earthquake along a lamination λ with transverse measure µ . Then there is an isometry I from the hyperbolic plane to the future component of the convex hull of the graph of the boundary value homeomorphism φ of E .”

In the holomorphic description of projective structures, each fiber C^{3g-3} is mapped injectively by the holonomy to Hom(π1F, PSL2C)/PSL2C.

“However in the holomorphic description of projective structures, each fiber C 3 g -3 is mapped injectively by the holonomy to Hom ( π 1 F, PSL 2 C ) / PSL 2 C [36].”

• cites [36] Kerckhoff S.P. (1983). The Nielsen realization problem. Ann. Math. (2) 117: 235–265 sourceNot checked — no full text of the source

The length of the shear along the lamination is log s.

“The length of the shear along the lamination is log s .”

The examples can be inserted into the ruled part of the boundary of a domain of dependence, yielding manifolds that contain closed timelike curves.

“These examples can be 'inserted' into the ruled part of the boundary of a domain of dependence, yielding manifolds that contain closed timelike curves.”

Suppose A,B generate a linear group which is not virtually solvable and suppose that both A,B have infinite order. Then for all sufficiently large N, A^N and B^N freely generate a free group.

“Lemma 8. Suppose A,B generate a linear group which is not virtually solvable and suppose that both A,B have infinite order. Then for all sufficiently large N , A N and B N freely generate a free group.”

Supposing there is a plane disjoint from H(A), H(A) is the intersection of all the closed half-spaces with boundary a totally geodesic null plane which contain A.

“Supposing there is a plane disjoint from H ( A ), H ( A ) is the intersection of all the closed half-spaces with boundary a totally geodesic null plane which contain A .”

The shearing lamination and measure are naturally associated to the bending lamination and bending measure of the boundary of the convex hull in anti de Sitter space of the graph of φ regarded as a subset of Q.

“The shearing lamination and measure are naturally associated to the bending lamination and bending measure of the boundary of the convex hull in anti de Sitter space of the graph of φ regarded as a subset of Q .”

There does not exist a complete flat spacetime M with fundamental group π1M ≅ π1S where S is a closed surface of negative Euler characteristic.

“Proposition 25. There does not exist a complete flat spacetime M with fundamental group π 1 M ∼ = π 1 S where S is a closed surface of negative Euler characteristic.”

Any complete spacelike surface in anti de Sitter space which is a boundary component of the convex hull of a nowhere timelike curve in the boundary can be approximated by one with a discrete bending lamination.

“Any complete spacelike surface in anti de Sitter space which is a boundary component of the convex hull of a nowhere timelike curve in the boundary can be approximated by one with a discrete bending lamination”

There is also a unique right earthquake obtained from the past boundary of the convex hull.

“There is also a unique right earthquake obtained from the past boundary of the convex hull.”

If the holonomy of M fixes a point or the corresponding geodesic lamination has no isolated leaves, then M is a maximal spacetime.

“In particular, if the holonomy of M fi xes a point or the corresponding geodesic lamination has no isolated leaves, M is a maximal spacetime.”

H(A)=H(A∞).

“Extending the definitions of F ( D ) , P ( D ) , H ( D ) given in lemma 7 and its proof to the case of a noncompact spacelike surface, we see that H ( A ) = H ( A ∞ ) where H ( A ∞ ) has just been defined.”

The map φ ↦ E(φ) is equivariant with respect to pre-composition and post-composition with Iso+ H2.

“The map φ ↦→ E ( φ ) is equivariant with respect to pre-composition and post-composition with Iso + H 2 .”

The intersection A_r = A∩T_r is spacelike in the new Lorentzian structure.

“So the intersection A r = A ∩ T r is spacelike in the new Lorentzian structure.”

Given any measured geodesic lamination (λ',2μ) one can bend a spacelike geodesic plane along it, obtaining a complete convex surface A and the closure of A will be a disc in X∪Q with nowhere spacelike boundary.

“Given any measured geodesic lamination ( λ ′ , 2 µ ) one can bend a spacelike geodesic plane along it, obtaining a complete convex surface A and the closure of A will be a disc in X ∪ Q with nowhere spacelike boundary.”

It suffices to show pointwise convergence at one point in order to show that the sets A_r converge to a nowhere timelike circle.

“So it suffices to show pointwise convergence at one point in order to show that the sets A r converge to a nowhere timelike circle.”

The universal cover of either boundary component of Z is one of the two boundary components of X(φ).

“It follows that the universal cover of either boundary component of Z is one of the two boundary components of X ( φ ).”

If L has no isolated leaves, f(L+) is the universal cover of the new spacetime.

“and, if L has no isolated leaves, f ( L + ) is the universal cover of our new spacetime.”

cl A = A∩∂A is a disc.

“Now to see that cl A = A ∩ ∂A is a disc, consider the action of the diagonal subgroup of SO (2) × SO (2) on X ∩ Q . It gives a continuous bijection of cl A onto the disc. Since cl A is compact this is a homeomorphism.”

Since the leaves of ̃L are disjoint planes, Lemma 2 implies that <x(q+t(p-q)),x(q+t(p-q))> increases as t increases, strictly unless the path does not meet ̃L.

“since the leaves of ˜ L are disjoint planes, lemma 2 implies that 〈 x ( q + t ( p -q )) , x ( q + t ( p -q )) 〉 increases as t increases (and the increase is strict unless the path does not meet ˜ L ).”

• AssumptionRests on an assumption: The leaves of ˜L are disjoint planes.

Using the dual pairing and Poincaré duality, the bundle V is identified with the cotangent bundle to Teichmüller space.

“Using the dual pairing and Poincar´ e duality, this identifies the bundle V with the cotangent bundle to Teichm¨ uller space.”

The bundle V can also be identified with the tangent bundle to Teichmüller space.

“Of course we can also identify the bundle V with the tangent bundle to Teichm¨ uller space”

The linear holonomy of a complete flat spacetime is either solvable or else a discrete subgroup of O(2,1).

“Proposition 24. The linear holonomy of a complete flat spacetime is either solvable or else a discrete subgroup of O (2 , 1) .”

For each orientation preserving homeomorphism φ:S1→S1 there is a unique left earthquake E(φ):H2→H2 which extends continuously to H2∩S1 with value φ on S1.

“Proposition 22. For each orientation preserving homeomorphism φ : S 1 → S 1 there is a unique left earthquake E ( φ ) : H 2 → H 2 which extends continuously to H 2 ∩ S 1 with value φ on S 1 .”

As long as the earthquake is along a lamination which is a union of isolated leaves, the same argument is applicable.

“As long as the earthquake is along a lamination which is a union of isolated leaves, the same argument is applicable:”

If the hypersurface is a torus then the projective structure is an affine structure.

“If the hypersurface is a torus then the projective structure is an affine structure; see [36] for a discussion of affine structure on tori.”

• cites [36] Kerckhoff S.P. (1983). The Nielsen realization problem. Ann. Math. (2) 117: 235–265 sourceNot checked — no full text of the source

In general given a hyperbolic surface S and an earthquake on S determined by a measured geodesic lamination, the quotient of the top of the convex hull in de Sitter space of the graph on the quadric Q of the conjugating homeomorphism determined by the earthquake on S is isometric to the hyperbolic surface obtained by an earthquake with half as much transverse measure.

“so in general given a hyperbolic surface S and an earthquake on S determined by a measured geodesic lamination, the quotient of the top of the convex hull in de Sitter space of the graph on the quadric Q of the conjugating homeomorphism determined by the earthquake on S is isometric to the hyperbolic surface obtained by an earthquake with half as much transverse measure.”

int(H(F(φ)))/π1F is a domain of dependence which contains all other domains of dependence with the same holonomy.

“So int ( H ( F ( φ )) ) /π 1 F is a domain of dependence which contains all other domains of dependence with the same holonomy.”

If N(F) is a domain of dependence then N(F) embeds in int(H(F(φ)))/π1F.

“If N ( F ) is a domain of dependence then N ( F ) embeds in int ( H ( F ( φ )) ) /π 1 F .”

F has negative Euler characteristic.

“First we show that F has negative Euler characteristic.”

• AssumptionThe authors state this as an assumption: In the proof that F has negative Euler characteristic, F is assumed to be a torus.

There is an earthquake E:H2→H2 with shearing lamination given by the single geodesic b=c=0 and transverse measure log s.

“There is an earthquake E : H 2 → H 2 with shearing lamination given by the single geodesic b = c = 0 and transverse measure log s .”

The closed set H(A∞) contains all the sets H(Ar).

“Then the closed set H ( A ∞ ) contains all the sets H ( A r ).”

Let F be the earthquake with shearing lamination λ and transverse measure 1/2 μ. Then I∘F has as boundary value the map which sends p∈∂H2 to the corresponding point on the graph of φ, i.e., the intersection of the line in the left ruling through p with the graph.

“Let F be the earthquake with shearing lamination λ and transverse measure 1 2 µ . Then I ◦ F has as boundary value the map which sends p ∈ ∂ H 2 to the corresponding point on the graph of φ , i.e., the intersection of the line in the left ruling through p with the graph.”

∂A∞ is a nowhere timelike topological circle.

“It follows that ∂A ∞ is a nowhere timelike topological circle.”

The boundary value of the earthquake is the homeomorphism of RP1 which is the identity on negative reals and multiplication by s on positive reals.

“The boundary value of the earthquake is the homeomorphism of RP 1 which is the identity on negative reals and multiplication by s on positive reals.”

The intersection of A with any indefinite plane has closure an arc.

“It is easy to see that the intersection of A with any indefinite plane has closure an arc, and the pointwise convergence follows.”

M̃=R2+1 contains the universal covers of M1 and M2, where M1, M2 respectively are the future, respectively the past, of a future, respectively past directed closed convex spacelike surface.

“˜ M = R 2+1 contains the universal covers of M 1 and M 2 , where M 1 , M 2 respectively are the future, respectively the past, of a future, respectively past directed closed convex spacelike surface.”

The spectrum of the Weil-Petersson Laplacian on Teichmüller space is expected to be bounded away from zero because the mapping class group is non-amenable.

“Because the mapping class group is non-amenable, we expect that the spectrum of the Weil-Petersson Laplacian on Teichm¨ uller space is bounded away from zero.”

• AssumptionRests on an assumption: The mapping class group is non-amenable.

One can deduce that a 3+1-dimensional spacetime which is a small neighbourhood of a closed spacelike hypersurface and has irreducible linear holonomy is a deformation, by a 1-cocycle with values in R^{3+1}, of the quotient of the positive (or negative) light cone by a group acting cocompactly on the hyperbolic 3-space of unit timelike directions.

“One can deduce that a 3 + 1-dimensional spacetime which is a small neighbourhood of a closed spacelike hypersurface and has irreducible linear holonomy is a deformation, by a 1-cocycle with values in R 3+1 , of the quotient of the positive (or negative) light cone by a group acting cocompactly on the hyperbolic 3-space of unit timelike directions.”

There is an earthquake E defined on H2 with shearing lamination (λ',μ); up to an isometry the image of λ' is the 'lamination' λ'' which is the union of the geodesics l''.

“It follows, using the previous proposition, that there is an earthquake E defined on H 2 with shearing lamination ( λ ′ , µ ); up to an isometry the image of λ ′ is the 'lamination' λ ′′ which is the union of the geodesics l ′′ .”

By the end of the proof of proposition 3, all these structures can be extended to complete flat Lorentz structures on M.

“By the end of the proof of proposition 3, all these structures can be extended to complete flat Lorentz structures on M .”

Kulkarni and Raymond [53] showed that a closed 3-manifold with a complete anti de Sitter structure is necessarily a Seifert manifold.

“Kulkarni and Raymond [53] showed that a closed 3-manifold with a complete anti de Sitter structure is necessarily a Seifert manifold.”

• cites [53]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Not all earthquakes have boundary values which are homeomorphisms.

“Indeed, not all earthquakes have boundary values which are homeomorphisms.”

The space of classical solutions to Einstein's equation in 2+1 dimensions on a manifold S × (0,1) containing S as a closed spacelike surface is parametrized by the cotangent space of the Teichmüller space of S.

“This family is parametrized by the cotangent space of the Teichm¨ uller space of S”

Since π1M has no normal abelian subgroup, the linear holonomy injects π1M into SO(2,1)0.

“Since π 1 M has no normal abelian subgroup, the linear holonomy injects π 1 M into SO (2 , 1) 0 .”

Since M is complete, π1M injects into ISO(2,1).

“Since M is complete, π 1 M injects into ISO (2 , 1).”

For a wave function in the Hilbert space, there is a uniform lower bound on the uncertainty of the translational part of the holonomy.

“This means that for a wave function in the Hilbert space, there is a lower bound on the uncertainty of the translational part of the holonomy, and unlike the case of Euclidean space the bound is uniform”

[54] shows that given a complete Lorentzian manifold of the form PSL2R/Γ where Γ is a cocompact subgroup acting by right translation, the deformations with left holonomy nontrivial but lying in an abelian subgroup are still complete.

“[54] shows that given a complete Lorentzian manifold of the form PSL 2 R / Γ where Γ is a cocompact subgroup acting by right translation, the deformations with left holonomy nontrivial but lying in an abelian subgroup are still complete.”

• cites [54]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Since the inclusion of either boundary component of M0 into M0 is a homotopy equivalence, M0 is compact.

“Since the inclusion of either boundary component of M 0 into M 0 is a homotopy equivalence, M 0 is compact.”

A covering of a compact Riemannian manifold with non-amenable deck group has spectrum bounded away from zero.

“A covering of a compact Riemannian manifold with non-amenable deck group has spectrum bounded away from zero, by Brooks's theorem [45]”

• cites [45] Fried D., Goldman W.M. (1983). Three-dimensional affine crystallographic groups. Adv. Math. 47(1): 1–49 sourceNot checked — no full text of the source

Given any nowhere spacelike circle in Q there is a corresponding left and right earthquake pair, each defined on the interior of a convex subset of the hyperbolic plane with geodesic boundary.

“Given any nowhere spacelike circle in Q there is a corresponding left and right earthquake pair, each defined on the interior of a convex subset of the hyperbolic plane with geodesic boundary.”

Each conformally null segment of a line in the left ruling in the boundary accounts for one geodesic on the frontier of the range of the left earthquake and one geodesic on the frontier of the domain of the right earthquake.

“Each conformally null segment of a line in the left ruling in the boundary accounts for one geodesic on the frontier of the range of the left earthquake and one geodesic on the frontier of the domain of the right earthquake.”

The closure cl A of the resulting convex surface meets Q in the limit set of a Fuchsian group which uniformizes a planar surface with three boundary circles, together with sawteeth.

“The closure cl A of the resulting convex surface meets Q in the limit set of a Fuchsian group which uniformizes a planar surface with three boundary circles, together with sawteeth.”

Even holding the linear holonomy fixed, there are arbitrarily small deformations of the holonomy which do not act fixed point free on R2+1 and therefore cannot be the deck groups of a covering of M by R2+1.

“But even holding the linear holonomy fixed, there are arbitrarily small deformations of the holonomy which do not act fixed point free on R 2+1 and therefore cannot be the deck groups of a covering of M by R 2+1 .”

Since the holonomy acts properly discontinuously on the neighbourhood of the surface, and therefore also on the restriction of the frame bundle to this neighbourhood, the holonomy of the surface must be discrete in the isometry group.

“Since the holonomy acts properly discontinuously on the neighbourhood of the surface, and therefore also on the restriction of the frame bundle to this neighbourhood, the holonomy of the surface must be discrete in the isometry group.”

• AssumptionRests on an assumption: The identity component of the isometry group of the model space acts simply transitively on the oriented orthochronous Lorentzian orthonormal frame bundle of the model space.

If M is a complete oriented orthochronous spacetime, then π1M is a discrete subgroup of ISO(2,1).

“So π 1 M is a discrete subgroup of ISO (2 , 1).”

If π1M is not solvable and the linear holonomy is not discrete, the linear holonomy must be a dense subgroup of SO(2,1)0.

“If π 1 M is not solvable and the linear holonomy is not discrete, the linear holonomy must be a dense subgroup of SO (2 , 1) 0 .”

By Koszul (proposition 1.3.2 of [5]), M̂' is E_x, regarded as an open subset of R^{2+1}.

“By a result of Koszul (proposition 1.3.2 of [5]), ̂ M ′ is E x , regarded as an open subset of R 2+1 .”

• cites [5] Witten, E.: On the structure of the topological phase of two dimensional quantum gravity. Preprint IASSNS-HEP-89/66, Institute for Advanced StudyNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The top surface of the convex hull determines a left earthquake on the interior of the convex core of the planar surface.

“The top surface of the convex hull determines a left earthquake on the interior of the convex core of the planar surface.”

The cotangent space of the Teichmüller space of S has a natural symplectic structure.

“the cotangent space has a natural symplectic structure.”

The earthquake with half as much transverse measure turns the surface with geodesic boundary into a thrice punctured sphere.

“The earthquake with half as much transverse measure turns the surface with geodesic boundary into a thrice punctured sphere, an example of proposition 23.”

By Jørgensen's inequality, if every 2-generator subgroup of a finitely generated subgroup H of SO(2,1)0 is discrete, H is discrete.

“Furthermore, by Jørgensen's inequality, if every 2-generator subgroup of a finitely generated subgroup H of SO (2 , 1) 0 is discrete, H is discrete.”

Proposition 27 establishes Goldman's conjecture that the deformations in [54] are complete.

“Goldman conjectured in [54] that these manifolds would still be complete; this is established by proposition 27.”

• cites [54]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The trace of the linear holonomy of an element of the fundamental group becomes an observable.

“In particular the trace of the linear holonomy of an element of the fundamental group becomes an observable.”

For each geodesic l' in λ', there are corresponding geodesics l and l'' in H2=X∩P obtained by left and respectively right translation of the endpoints.

“For each geodesic l ′ in λ ′ , there are corresponding geodesics l and l ′′ in H 2 = X ∩ P obtained by left and respectively right translation of the endpoints.”

The graph in Q of f:R→R, t↦e^t union a single segment of a null line has a convex hull, whose boundary components give earthquakes taking a complete hyperbolic surface with cyclic fundamental group and one cusp to one of the components of the complement of a closed geodesic in a hyperbolic surface with cyclic fundamental group and no cusp.

“The graph in Q of f : R → R , t ↦→ e t union a single segment of a null line has a convex hull, whose boundary components give earthquakes taking a complete hyperbolic surface with cyclic fundamental group and one cusp to one of the components of the complement of a closed geodesic in a hyperbolic surface with cyclic fundamental group and no cusp.”