Drilling, Pants Decompositions and
Ending Laminations
Jeffrey Brock
Brown University
Drilling, Pants Decompositions andEnding Laminations – p.1/36
Joint work with
K. Bromberg – Utah
R. Evans – Auckland
J. Souto – Chicago and Michigan
Drilling, Pants Decompositions andEnding Laminations – p.2/36
Main Goal
To give a new streamlined perspective on the
ending lamination theorem:
Ending Lamination Theorem. (B-C-M) A tame
hyperbolic 3-manifold is determined by its topology, its
cusps, and its end invariants.
Drilling, Pants Decompositions andEnding Laminations – p.3/36
Philosophy
Original philosophy:
Show each M with end-invariant ν is uniformly
bi-Lipschitz to a model manifold Mν .
Apply Sullivan’s rigidity theorem.
Drilling, Pants Decompositions andEnding Laminations – p.4/36
Philosophy
Alternate philosophy:
Show that each M with end-invariant ν is a limit of Mn
canonically determined by ν .
Can we generate approximates Mn directly from
combinatorial information about ν ?
Drilling, Pants Decompositions andEnding Laminations – p.5/36
Specialize to a Bers Slice
Simplifying assumptions: our manifolds Q will
lie in the boundary of a Bers slice:
BX = {Q(X,Y ) | Y ∈ Teich(S)}.
Given Q ∈ ∂ BX with no non-peripheral cusps,
the ending lamination conjecture predicts that Q
is determined by its ending lamination ν .
ν is a limit of simple closed non-peripheral
curves on S whose geodesic representatives exit
the convex core of Q.
Drilling, Pants Decompositions andEnding Laminations – p.6/36
The Punctured Torus (Minsky)
When T is a punctured torus, π1 (T ) = ha, bi,
require ρ ([a, b]) to be parabolic.
The end-invariant ν lies in R ∪ ∞.
ν ∈ Q ∪ ∞ represents a cusp, ν ∈ R \ Q represent
laminations of “slope” ν .
Drilling, Pants Decompositions andEnding Laminations – p.7/36
The Punctured Torus
The lift of a lamination on the punctured torus to H2 .
Drilling, Pants Decompositions andEnding Laminations – p.8/36
Laminations
The end-invariant ν will in general be a geodesic
lamination.
Definition. A geodesic lamination on a hyperbolic
surface X is a closed subset foliated by geodesics.
Drilling, Pants Decompositions andEnding Laminations – p.9/36
The Farey Graph
∞
3
−3
2
−2
√
1+ 5
2
1
−1
−1/2
1/2
−1/3
1/3
0
Drilling, Pants Decompositions andEnding Laminations – p.10/36
Continued Fractions
The Farey graph encodes the continued fraction
expansion of ν .
The partial quotients qn are the pivots in the
Farey sequence – qn approaches ν .
A priori bounds: A theorem of Minsky guarantees
that the curves whose slopes correspond to pivots qn
have bounded length in Q.
Any limit Q∞ of the manifolds Cn with cusps at
qn satisfies ν (Q∞ ) = ν . Suffices to show that Cn
converges to Q.
Drilling, Pants Decompositions andEnding Laminations – p.11/36
Cusps are Dense
A maximal cusp MC(P) in ∂ BX is uniquely
determined by a pants decomposition P of S
whose elements are all cusps in MC(P).
Theorem. (McMullen) Maximal cusps are dense in
the Bers boundary ∂ BX .
How do rational points encode irrational ones?
Finding a canonical family of maximal cusps
becomes a combinatorial problem...
Drilling, Pants Decompositions andEnding Laminations – p.12/36
Higher Farey graphs
Given ν , work of Masur and Minsky, and Klarreich,
the complex of curves C (S) produces a sequence
{Pn } → ν of pants decompositions of S via its
hierarchical structure.
Theorem. (Minsky – a priori bounds). In any
Q ∈ ∂ BX with ending lamination ν , the pants
decompositions Pn have uniformly bounded length.
Brian Bowditch has developed a new perspective
and proof of a priori bounds and the model.
Drilling, Pants Decompositions andEnding Laminations – p.13/36
A priori approximates
Theorem. (B-Bromberg-Evans-Souto) Given Q
with ending lamination ν , the maximal cusps
MC(Pn ) ∈ ∂ BX corresponding to canonical pants
decompositions Pn → ν converge to Q.
Corollary. (ELC) The manifold Q is uniquely
determined by ν .
An earlier result due to R. Evans treats the case
that ν predicts short pants decompositions in Q.
Drilling, Pants Decompositions andEnding Laminations – p.14/36
Approximations by cusps
The proof of the main theorem relies on the
following approximation theorem:
Theorem (BBES). For all B > BS there exists L > 1 so
that the following holds: Let Q ∈ ∂ BX have ending
lamination ν and let {Pn } → ν satisfy `Q (Pn ) < B.
e ∈ ∂ B e so that for each
Then there is a finite cover Q
X
e there are, for n 0, L-bi-Lipschitz
compact set K ⊂ Q
φn : K ,→ MC(Pen ).
Drilling, Pants Decompositions andEnding Laminations – p.15/36
Two Pieces
I. Describe how to obtain Pn .
II. Show convergence of MC(Pn ) to Q using the
approximation theorem.
Drilling, Pants Decompositions andEnding Laminations – p.16/36
Higher Continued Fractions
To find Pn one employs generalized continued
fractions.
Each essential subsurface of S inherits a
projection coefficient via the geometry of the
complex of curves.
Work of Masur and Minsky organizes these data
into a hierarchy Hν of geodesics in curve
complexes C (T ), T ⊆ S.
Hν determines useful sequences of pants
decompositions depending only on ν .
Drilling, Pants Decompositions andEnding Laminations – p.17/36
The Complex of Curves
The complex of curves C (S) organizes the
essential simple closed curves on S.
0-skeleton: isotopy classes of essential simple
closed curves on S.
k-simplices: spanned by families of k + 1 curves
α0 , . . . , αk with i(αi , α j ) = 0.
Drilling, Pants Decompositions andEnding Laminations – p.18/36
The Complex of Curves
Simplices in the complex of curves.
Drilling, Pants Decompositions andEnding Laminations – p.19/36
Exceptional case
When S is a one-holed torus or four-holed
sphere, edges connect vertices whose curves
intersect minimally.
Theorem. (Masur-Minsky) In each case, C (S) is
negatively curved in the sense of Gromov.
Theorem. (Klarreich) ∂ C (S) = E L (S).
Drilling, Pants Decompositions andEnding Laminations – p.20/36
Hierarchies
Masur-Minsky exhibit a geodesic gν in C (S)
associated to ν joining the shortest curve γX on
X to ν ∈ ∂ C (S).
A hierarchy Hν of geodesics in C (T ), T ⊂ S,
gives a kind of thickening of gν in C (S).
Drilling, Pants Decompositions andEnding Laminations – p.21/36
Hierarchies
α−1
β1
α1
α0
β2
A hierarchy in the two-holed torus case
Drilling, Pants Decompositions andEnding Laminations – p.22/36
Hierarchies and Pants
β1
β2
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Hierarchies and Pants
β1
β2
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Hierarchies and Pants
β1
β2
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Hierarchies and Pants
β1
β2
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Hierarchies and Pants
β1
β2
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Hierarchies and Pants
β1
β2
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Hierarchies and Pants
β1
β2
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Hierarchies and Pants
β1
β2
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Hierarchies and Pants
β1
β2
The spokes of the wheels are pants decompositions.
Drilling, Pants Decompositions andEnding Laminations – p.23/36
Subsurface Projections
Roughly speaking, the laminations determine curves
in H via a surgery procedure: subsurface projection.
Drilling, Pants Decompositions andEnding Laminations – p.24/36
Subsurface Projections
Roughly speaking, the laminations determine curves
in H via a surgery procedure: subsurface projection.
Drilling, Pants Decompositions andEnding Laminations – p.24/36
Subsurface Projections
Roughly speaking, the laminations determine curves
in H via a surgery procedure: subsurface projection.
Drilling, Pants Decompositions andEnding Laminations – p.24/36
Subsurface Projections
Roughly speaking, the laminations determine curves
in H via a surgery procedure: subsurface projection.
Drilling, Pants Decompositions andEnding Laminations – p.24/36
Subsurface Projections
Roughly speaking, the laminations determine curves
in H via a surgery procedure: subsurface projection.
Drilling, Pants Decompositions andEnding Laminations – p.24/36
Subsurface Projections
Roughly speaking, the laminations determine curves
in H via a surgery procedure: subsurface projection.
Drilling, Pants Decompositions andEnding Laminations – p.24/36
Subsurface Projections
Roughly speaking, the laminations determine curves
in H via a surgery procedure: subsurface projection.
Drilling, Pants Decompositions andEnding Laminations – p.24/36
Subsurface Projections
Roughly speaking, the laminations determine curves
in H via a surgery procedure: subsurface projection.
The result is the projection πY (ν ) where Y ⊂ S and
ν is the ending lamination.
Drilling, Pants Decompositions andEnding Laminations – p.24/36
Structure of Hν
α−1
α0
α1
Here, if Y is the one-holed torus bounded by α0 ,
α−1 ∼ πY (γX )
and
α1 ∼ πY (ν ).
Drilling, Pants Decompositions andEnding Laminations – p.25/36
Convergence
It remains to show
lim MC(Pn ) = Q.
n→∞
Theorem (BBES). For all B > BS there exists L > 1 so
that the following holds: Let Q ∈ ∂ BX have ending
lamination ν and let {Pn } → ν satisfy `Q (Pn ) < B.
e ∈ ∂ B e so that for each
Then there is a finite cover Q
X
e there are, for n 0, L-bi-Lipschitz
compact set K ⊂ Q
φn : K ,→ MC(Pen ).
Drilling, Pants Decompositions andEnding Laminations – p.26/36
Grafting and Drilling
Let shortε (Pn ) denote the curves α ∈ Pn for
which `Q (α ) < ε .
Step I. Graft and Drill geodesic representatives
of shortε (Pn ) out of Q, let Q0 be the
π1 (X)-cover.
Step II. Obtain Pn0 by < D moves rel-cusps on Pn
so that lifts of remaining α ∈ Pn0 may be drilled
e0 to obtain MC(Pen0 ) in ∂ B e .
in a finite cover Q
X
Drilling, Pants Decompositions andEnding Laminations – p.27/36
Moves are Bounded
Step III. Show that if P and P0 differ by an
elementary move then MC(P) and MC(P0 ) are
uniformly bi-Lipschitz on large subsets.
e by
Step IV. The covers MC(Pen0 ) converge to Q
Sullivan’s rigidity theorem, so by Step III,
lim MC(Pn ) = lim MC(Pn0 ) = Q.
Drilling, Pants Decompositions andEnding Laminations – p.28/36
Step I.
We comment on aspects of these steps.
Step I requires an application of the method of
grafting (Bromberg), and an application of the
drilling theorem:
Theorem (B-Bromberg). Let M be a geometrically
finite cone-manifold. For all K > 1 there exists an
ε > 0 so that if c is a closed geodesic in M with
`M (c) < ε , then there is a K-bi-Lipschitz diffeo.
φ : M \Uc → M0 \ Pc .
Drilling, Pants Decompositions andEnding Laminations – p.30/36
Step II – Almost Convex
A priori bounds guarantees each remaining
α ∈ Pn satisfies `Q0 (α ) < BS .
Realizing remaining curves by pleated surfaces
gives a collection of thick subsurfaces Z in Q0 .
[Almost Convex] If Z is homotopically near
∂ CC(Q0 ), we can pinch Pn ∩ Z by
quasi-conformal deformation.
Otherwise there is a nearby thin part, or Z lies
in a thick product region homeomorphic to
Z × I.
Drilling, Pants Decompositions andEnding Laminations – p.31/36
Step II – Short Nearby
[Short Nearby] If there is a nearby Margulis
tube, a geometric limit argument together with
a theorem of Otal shows there is a nearby
Margulis tube Tγ so that γ ∈ C (S), and
`Z (γ ) < L0 .
Can extend γ to a bounded length pants
decomposition Pγ on Z, which thus has bounded
distance from Pn ∩ Z – drill γ and reduce.
Drilling, Pants Decompositions andEnding Laminations – p.32/36
Step II – Thick Product
[Thick Product] If Z lies in a thick product
region, we may apply a finite cover trick
(Bromberg-Souto) to drill a longer curve using
recent innovations of Hodgson-Kerckhoff.
Idea: To drill α , pass to a finite cover in which α
is unknotted and has a large embedded tube.
Then the grafting/cone-deformation
construction works in the cover.
By a geometric limit argument, can reduce by
pinching all translates of α .
Drilling, Pants Decompositions andEnding Laminations – p.33/36
Step III – Moves Bounded.
[Moves Bounded] If we perform an elementary
move P → P0 , we change exactly one cusp α ∈ P
to β ∈ P0 with i(α , β ) = 1 or 2.
There is a single almost maximal cusp Qα ,β
where P \ α is pinched, and α and β each satisfy
`∂ Qα ,β (α ) < K and `∂ Qα ,β (β ) < K.
Large subsets of MC(P) and MC(P0 ) embed in
Qα ,β with uniformly bounded bi-Lipschitz
constant.
Drilling, Pants Decompositions andEnding Laminations – p.34/36
Step V – conclusion
e uniformly
Large compact subsets of Q
bi-Lipschitz embed into MC(Pen0 ) for n 0.
e∞ of MC(Pen0 ) is
Diagonalizing, any limit Q
e
uniformly bi-Lipschitz to Q.
Sullivan rigidity guarantees
e∞ = Q
e so Q∞ = Q.
Q
Arguing similarly, [Moves-Bounded] gives:
lim MC(Pn ) = Q.
Drilling, Pants Decompositions andEnding Laminations – p.35/36
Next time...
Use surfaces to control 3-manifolds
fiberings over S1
Heegaard splittings
Drilling, Pants Decompositions andEnding Laminations – p.36/36