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(1)

Virtually infinite first Betti number.

Finite covers of a hyperbolic 3-manifold and virtual fibers.

Claire Renard

Institut de Math ´ematiques de Toulouse

November 2nd 2011

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(2)

Virtually infinite first Betti number.

Some conjectures.

Let M be a hyperbolic 3-manifold, connected, closed and oriented.

Theorem (Kahn, Markovic)

The fundamental group π 1 M contains a surface subgroup.

Conjectures

(1) (Virtually Haken.) There exists a finite cover M 0 → M containing an

incompressible surface, i.e. an embedded surface T in M 0 such that the map induced by the embedding ι : T → M 0 on fundamental groups ι ∗ : π 1 T → π 1 M 0 is injective.

(2) (Virtually positive first Betti number.) There exists a finite cover M 0 → M with b 1 (M 0 ) > 0.

(3) (Virtually infinite first Betti number.) For each n ∈ N, there exists a finite cover M n → M with b 1 (M n ) ≥ n.

(4) (Thurston.) There exists a finite cover M 0 → M which fibers over the circle S 1 .

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(3)

Virtually infinite first Betti number.

Some conjectures.

Let M be a hyperbolic 3-manifold, connected, closed and oriented.

Theorem (Kahn, Markovic)

The fundamental group π 1 M contains a surface subgroup.

Conjectures

(1) (Virtually Haken.) There exists a finite cover M 0 → M containing an

incompressible surface, i.e. an embedded surface T in M 0 such that the map induced by the embedding ι : T → M 0 on fundamental groups ι ∗ : π 1 T → π 1 M 0 is injective.

(2) (Virtually positive first Betti number.) There exists a finite cover M 0 → M with b 1 (M 0 ) > 0.

(3) (Virtually infinite first Betti number.) For each n ∈ N, there exists a finite cover M n → M with b 1 (M n ) ≥ n.

(4) (Thurston.) There exists a finite cover M 0 → M which fibers over the circle S 1 .

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(4)

Virtually infinite first Betti number.

Some conjectures.

Let M be a hyperbolic 3-manifold, connected, closed and oriented.

Theorem (Kahn, Markovic)

The fundamental group π 1 M contains a surface subgroup.

Conjectures

(1) (Virtually Haken.) There exists a finite cover M 0 → M containing an

incompressible surface, i.e. an embedded surface T in M 0 such that the map induced by the embedding ι : T → M 0 on fundamental groups ι ∗ : π 1 T → π 1 M 0 is injective.

(2) (Virtually positive first Betti number.) There exists a finite cover M 0 → M with b 1 (M 0 ) > 0.

(3) (Virtually infinite first Betti number.) For each n ∈ N, there exists a finite cover M n → M with b 1 (M n ) ≥ n.

(4) (Thurston.) There exists a finite cover M 0 → M which fibers over the circle S 1 .

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(5)

Virtually infinite first Betti number.

Some conjectures.

Let M be a hyperbolic 3-manifold, connected, closed and oriented.

Theorem (Kahn, Markovic)

The fundamental group π 1 M contains a surface subgroup.

Conjectures

(1) (Virtually Haken.) There exists a finite cover M 0 → M containing an

incompressible surface, i.e. an embedded surface T in M 0 such that the map induced by the embedding ι : T → M 0 on fundamental groups ι ∗ : π 1 T → π 1 M 0 is injective.

(2) (Virtually positive first Betti number.) There exists a finite cover M 0 → M with b 1 (M 0 ) > 0.

(3) (Virtually infinite first Betti number.) For each n ∈ N, there exists a finite cover M n → M with b 1 (M n ) ≥ n.

(4) (Thurston.) There exists a finite cover M 0 → M which fibers over the circle S 1 .

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(6)

Virtually infinite first Betti number.

Some conjectures.

Let M be a hyperbolic 3-manifold, connected, closed and oriented.

Theorem (Kahn, Markovic)

The fundamental group π 1 M contains a surface subgroup.

Conjectures

(1) (Virtually Haken.) There exists a finite cover M 0 → M containing an

incompressible surface, i.e. an embedded surface T in M 0 such that the map induced by the embedding ι : T → M 0 on fundamental groups ι ∗ : π 1 T → π 1 M 0 is injective.

(2) (Virtually positive first Betti number.) There exists a finite cover M 0 → M with b 1 (M 0 ) > 0.

(3) (Virtually infinite first Betti number.) For each n ∈ N, there exists a finite cover M n → M with b 1 (M n ) ≥ n.

(4) (Thurston.) There exists a finite cover M 0 → M which fibers over the circle S 1 .

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(7)

Virtually infinite first Betti number.

Some conjectures.

Let M be a hyperbolic 3-manifold, connected, closed and oriented.

Theorem (Kahn, Markovic)

The fundamental group π 1 M contains a surface subgroup.

Conjectures

(1) (Virtually Haken.) There exists a finite cover M 0 → M containing an

incompressible surface, i.e. an embedded surface T in M 0 such that the map induced by the embedding ι : T → M 0 on fundamental groups ι ∗ : π 1 T → π 1 M 0 is injective.

(2) (Virtually positive first Betti number.) There exists a finite cover M 0 → M with b 1 (M 0 ) > 0.

(3) (Virtually infinite first Betti number.) For each n ∈ N, there exists a finite cover M n → M with b 1 (M n ) ≥ n.

(4) (Thurston.) There exists a finite cover M 0 → M which fibers over the circle S 1 .

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(8)

Virtually infinite first Betti number.

Definition

A 3-manifold M is said to be virtually fibered if it admits a finite cover which fibers over the circle.

An embedded surface S in M is a virtual fiber if there is a finite cover of M in which the preimage of S is a fiber.

T {1/2}

T {0} T {1}

M’= T I /(x,0) ~ ( x,1)ϕ

M ϕ

S

Question: Let M 0 → M be a finite cover of M. Find conditions for M 0 to contain an embedded surface which is a fiber, or at least a virtual fiber ?

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(9)

Virtually infinite first Betti number.

Definition

A 3-manifold M is said to be virtually fibered if it admits a finite cover which fibers over the circle.

An embedded surface S in M is a virtual fiber if there is a finite cover of M in which the preimage of S is a fiber.

T {1/2}

T {0} T {1}

M’= T I /(x,0) ~ ( x,1)ϕ

M ϕ

S

Question: Let M 0 → M be a finite cover of M. Find conditions for M 0 to contain an embedded surface which is a fiber, or at least a virtual fiber ?

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(10)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1

Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2

k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(11)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1

Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2

k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(12)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1

Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2

k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(13)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1

Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2

k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(14)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2

k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(15)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2 k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(16)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2 k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(17)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2 k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(18)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2 k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(19)

Virtually infinite first Betti number.

Main theorem.

Theorem (1, main theorem.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k = k (, Vol(M)) such that:

If M 0 → M is a cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2 k g ln g < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 contains an embedded surface T of genus g(T ) ≤ g which is a virtual fiber.

In particular, M virtually fibers over the circle and M 0 is Haken.

Conjecture (∗)

The technical assumption (1) is not necessary.

Remark

If Vol(M) is fixed, lim →0 k (, Vol(M)) → +∞.

If  is fixed, lim Vol(M)→+∞ k (, Vol(M)) → +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(20)

Virtually infinite first Betti number.

Ideas of the proof of the main theorem.

Suppose that the ratio g ln g/ ln ln(d /q) is “small enough”.

Proof in two steps.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(21)

Virtually infinite first Betti number.

First step: Construct an embedded “long and thin” product T × [0, m] in M 0 , satisfying the following properties.

T T T T

T [0,m]

M’

0 1 2

T

m−1 m

< K(g)

> r

The surface T is orientable and closed, with genus g(T ) ≤ g.

The number m = m( d q , g) is “large”.

The surfaces T j := T × {j} have their diameters uniformly bounded from above by K = K (g).

Two surfaces T j and T j+1 are at distance at least r > 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(22)

Virtually infinite first Betti number.

First step: Construct an embedded “long and thin” product T × [0, m] in M 0 , satisfying the following properties.

T T T T

T [0,m]

M’

0 1 2

T

m−1 m

< K(g)

> r

The surface T is orientable and closed, with genus g(T ) ≤ g.

The number m = m( d q , g) is “large”.

The surfaces T j := T × {j} have their diameters uniformly bounded from above by K = K (g).

Two surfaces T j and T j+1 are at distance at least r > 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(23)

Virtually infinite first Betti number.

Second step: Use this product to construct a virtual fibration of M 0 .

T ’1 P2

T1 T T

T

0 2 3

P1

Choose D, a Dirichlet fundamental polyhedron for M in H 3 .

For each surface T j , consider the pattern of fundamental domains P j that is the union of the fundamental domains meeting T j .

Find two patterns:

1 disjoint from each other,

2 isometric to the same ”model” pattern P,

3 containing parallel surfaces T

1

and T

10

.

Cut along T 1 and T 1 0 , glue them together to get a finite fibered cover N of M.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(24)

Virtually infinite first Betti number.

Second step: Use this product to construct a virtual fibration of M 0 .

T ’1 P2

T1 T T

T

0 2 3

P1

Choose D, a Dirichlet fundamental polyhedron for M in H 3 .

For each surface T j , consider the pattern of fundamental domains P j that is the union of the fundamental domains meeting T j .

Find two patterns:

1 disjoint from each other,

2 isometric to the same ”model” pattern P,

3 containing parallel surfaces T

1

and T

10

.

Cut along T 1 and T 1 0 , glue them together to get a finite fibered cover N of M.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(25)

Virtually infinite first Betti number.

Second step: Use this product to construct a virtual fibration of M 0 .

T ’1 P2

T1 T T

T

0 2 3

P1

Choose D, a Dirichlet fundamental polyhedron for M in H 3 .

For each surface T j , consider the pattern of fundamental domains P j that is the union of the fundamental domains meeting T j .

Find two patterns:

1 disjoint from each other,

2 isometric to the same ”model” pattern P,

3 containing parallel surfaces T

1

and T

10

.

Cut along T 1 and T 1 0 , glue them together to get a finite fibered cover N of M.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(26)

Virtually infinite first Betti number.

Second step: Use this product to construct a virtual fibration of M 0 .

T ’1 P2

T1 T T

T

0 2 3

P1

Choose D, a Dirichlet fundamental polyhedron for M in H 3 .

For each surface T j , consider the pattern of fundamental domains P j that is the union of the fundamental domains meeting T j .

Find two patterns:

1 disjoint from each other,

2 isometric to the same ”model” pattern P,

3 containing parallel surfaces T

1

and T

10

.

Cut along T 1 and T 1 0 , glue them together to get a finite fibered cover N of M.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(27)

Virtually infinite first Betti number.

Second step: Use this product to construct a virtual fibration of M 0 .

T ’1 P2

T1 T T

T

0 2 3

P1

Choose D, a Dirichlet fundamental polyhedron for M in H 3 .

For each surface T j , consider the pattern of fundamental domains P j that is the union of the fundamental domains meeting T j .

Find two patterns:

1 disjoint from each other,

2 isometric to the same ”model” pattern P,

3 containing parallel surfaces T

1

and T

10

.

Cut along T 1 and T 1 0 , glue them together to get a finite fibered cover N of M.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(28)

Virtually infinite first Betti number.

P2

ϕ2

T1 T1

T T

T

0 2 3

ϕ1

ψ

T P P1

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(29)

Virtually infinite first Betti number.

?

?

×

×

T 1 0 T 1

M 0

M

× ?

N S 1

T b 1 S 1

F

W

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(30)

Virtually infinite first Betti number.

x

1 γ

γ4 γ3 γ2 1

γ0

x x x

x

x5 4 3 2

0

T’1 E+

E− E+

2 E−

2 P2

ϕ2 E+

E− P1

T T

T

0 2 3

1 1

T1

ϕ1 ψ

P

T

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(31)

Virtually infinite first Betti number.

The regular case.

Theorem (2, regular case.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k 0 = k 0 (, Vol(M)) such that:

If M 0 → M is a regular cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1

Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2

k 0 g 2 < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 is a fiber bundle over the circle, and a fiber can be obtained from a component of F , possibly after some surgeries.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(32)

Virtually infinite first Betti number.

The regular case.

Theorem (2, regular case.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k 0 = k 0 (, Vol(M)) such that:

If M 0 → M is a regular cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1

Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2

k 0 g 2 < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 is a fiber bundle over the circle, and a fiber can be obtained from a component of F , possibly after some surgeries.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(33)

Virtually infinite first Betti number.

The regular case.

Theorem (2, regular case.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k 0 = k 0 (, Vol(M)) such that:

If M 0 → M is a regular cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2

k 0 g 2 < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 is a fiber bundle over the circle, and a fiber can be obtained from a component of F , possibly after some surgeries.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(34)

Virtually infinite first Betti number.

The regular case.

Theorem (2, regular case.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k 0 = k 0 (, Vol(M)) such that:

If M 0 → M is a regular cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2 k 0 g 2 < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 is a fiber bundle over the circle, and a fiber can be obtained from a component of F , possibly after some surgeries.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(35)

Virtually infinite first Betti number.

The regular case.

Theorem (2, regular case.)

Fix  ≤ Inj(M)/2. There exists an explicit constant k 0 = k 0 (, Vol(M)) such that:

If M 0 → M is a regular cover of finite degree d,

with an embedded, closed, orientable, pseudo-minimal surface F , which splits M 0 into q compression bodies C 1 , . . . , C q with the following properties:

1 Every simple closed curve embedded in C j of length ≤  is nul-homotopic in C j .

2 k 0 g 2 < ln ln d/q, with g = max j {g(C j )}.

Then the finite cover M 0 is a fiber bundle over the circle, and a fiber can be obtained from a component of F , possibly after some surgeries.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(36)

Virtually infinite first Betti number.

Application to Heegaard splittings.

H1 H2

ϕ

F F

If F is a surface of genus at least 1, χ (F ) = 2g(F ) − 2.

Definition

The Heegaard characteristic: χ h (M) = 2g(M) − 2.

The strong Heegaard characteristic: χ sh (M) = max F {χ (F )} where F is a strongly irreducible Heegaard surface for M.

Remark

If M 0 → M is a cover of finite degree d, χ h (M 0 ) ≤ d χ h (M).

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(37)

Virtually infinite first Betti number.

Application to Heegaard splittings.

H1 H2

ϕ

F F

If F is a surface of genus at least 1, χ (F ) = 2g(F ) − 2.

Definition

The Heegaard characteristic: χ h (M) = 2g(M) − 2.

The strong Heegaard characteristic: χ sh (M) = max F {χ (F )} where F is a strongly irreducible Heegaard surface for M.

Remark

If M 0 → M is a cover of finite degree d, χ h (M 0 ) ≤ d χ h (M).

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(38)

Virtually infinite first Betti number.

Application to Heegaard splittings.

H1 H2

ϕ

F F

If F is a surface of genus at least 1, χ (F ) = 2g(F ) − 2.

Definition

The Heegaard characteristic: χ h (M) = 2g(M) − 2.

The strong Heegaard characteristic: χ sh (M) = max F {χ (F )} where F is a strongly irreducible Heegaard surface for M.

Remark

If M 0 → M is a cover of finite degree d, χ h (M 0 ) ≤ d χ h (M).

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(39)

Virtually infinite first Betti number.

Heegaard gradient and conjectures of Lackenby.

Definition (Lackenby) Heegaard gradient:

h (M) = inf

i

( χ h (M i ) d i

) .

Strong Heegaard gradient:

sh (M) = inf

i

( χ sh (M i ) d i

) .

Conjecture (Lackenby)

(1) The Heegaard gradient of M is zero if and only if M virtually fibers over the circle.

(2) The strong Heegaard gradient is always strictly positive.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(40)

Virtually infinite first Betti number.

Heegaard gradient and conjectures of Lackenby.

Definition (Lackenby) Heegaard gradient:

h (M) = inf

i

( χ h (M i ) d i

) .

Strong Heegaard gradient:

sh (M) = inf

i

( χ sh (M i ) d i

) .

Conjecture (Lackenby)

(1) The Heegaard gradient of M is zero if and only if M virtually fibers over the circle.

(2) The strong Heegaard gradient is always strictly positive.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(41)

Virtually infinite first Betti number.

The sub-logarithmic version is true.

Definition Let η ∈ (0, 1).

η-sub-logarithmic Heegaard gradient:

h log,η (M) = inf

i

( χ h (M i ) (ln ln d i ) η

) .

Strong η-sub-logarithmic Heegaard gradient:

sh log,η (M) = inf

i

( χ sh (M i ) (ln ln d i ) η

) .

Proposition (3, Sub-logarithmic version of Lackenby’s conjectures.) Suppose conjecture (∗) is true.

Let η ∈ (0, 1).

(1) The η-sub-logarithmic Heegaard gradient of M is zero if and only if M virtually fibers over the circle.

(2) The strong η-sub-logarithmic Heegaard gradient is always strictly positive.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(42)

Virtually infinite first Betti number.

The sub-logarithmic version is true.

Definition Let η ∈ (0, 1).

η-sub-logarithmic Heegaard gradient:

h log,η (M) = inf

i

( χ h (M i ) (ln ln d i ) η

) .

Strong η-sub-logarithmic Heegaard gradient:

sh log,η (M) = inf

i

( χ sh (M i ) (ln ln d i ) η

) .

Proposition (3, Sub-logarithmic version of Lackenby’s conjectures.) Suppose conjecture (∗) is true.

Let η ∈ (0, 1).

(1) The η-sub-logarithmic Heegaard gradient of M is zero if and only if M virtually fibers over the circle.

(2) The strong η-sub-logarithmic Heegaard gradient is always strictly positive.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(43)

Virtually infinite first Betti number.

The homological viewpoint.

Let α ∈ H 1 ( M, Z) be a non-trivial element.

Definition

A kαk-minimizing surface R is an embedded surface with homology class Poincar ´e-dual to α, and minimizing Thurston’s norm: χ − (R) = kαk.

Question: Find conditions to ensure that R is the fiber of a fibration over the circle ?

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(44)

Virtually infinite first Betti number.

The homological viewpoint.

Let α ∈ H 1 ( M, Z) be a non-trivial element.

Definition

A kαk-minimizing surface R is an embedded surface with homology class Poincar ´e-dual to α, and minimizing Thurston’s norm: χ − (R) = kαk.

Question: Find conditions to ensure that R is the fiber of a fibration over the circle ?

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(45)

Virtually infinite first Betti number.

The homological viewpoint.

Let α ∈ H 1 ( M, Z) be a non-trivial element.

Definition

A kαk-minimizing surface R is an embedded surface with homology class Poincar ´e-dual to α, and minimizing Thurston’s norm: χ − (R) = kαk.

Question: Find conditions to ensure that R is the fiber of a fibration over the circle ?

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(46)

Virtually infinite first Betti number.

Circular decompositions.

Definition

Let M R be the 3-manifold obtained from M by removing a regular neighborhood of R diffeomorphic to R × (−1, 1). The circular characteristic of α, denoted by χ c (α), is the minimum over all kαk-minimizing surfaces R of the Heegaard characteristic of the cobordism (M R ,R × {1}, R × {−1}).

R {−1}

R {1}

MR

S1

1~(−1)

R f

M

S

Remark

χ c (α) = kαk + h(α), where h(α) is the minimum over all kαk-minimizing surfaces R of the minimal number of critical points of index 1 and 2 of a Morse function

M R → [−1, 1].

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(47)

Virtually infinite first Betti number.

Circular decompositions.

Definition

Let M R be the 3-manifold obtained from M by removing a regular neighborhood of R diffeomorphic to R × (−1, 1). The circular characteristic of α, denoted by χ c (α), is the minimum over all kαk-minimizing surfaces R of the Heegaard characteristic of the cobordism (M R ,R × {1}, R × {−1}).

R {−1}

R {1}

MR

S1

1~(−1)

R f

M

S

Remark

χ c (α) = kαk + h(α), where h(α) is the minimum over all kαk-minimizing surfaces R of the minimal number of critical points of index 1 and 2 of a Morse function

M R → [−1, 1].

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(48)

Virtually infinite first Betti number.

Applications to circular decompositions.

Theorem (4, Adapted from a result of Lackenby)

There exists an explicit constant ` 0 = ` 0 (, Vol(M)) such that:

Fix α ∈ H 1 (M) a non-trivial cohomology class and R a kαk-minimizing surface. Let M 0 → M be a d-sheeted regular cover and α 0 ∈ H 1 (M 0 , Z) the Poincar ´e-dual class associated to a connected component R 0 of the preimage of R in M 0 .

If ` 0 χ c 0 ) ≤ √

4

d, then the manifold M fibers over the circle and the surface R is a fiber.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(49)

Virtually infinite first Betti number.

Virtually infinite first Betti number.

Question: Find a tower of finite covers . . . → M i+1 → M i → . . . → M such that lim i→+∞ b 1 (M i ) = +∞ ?

Theorem (5)

Suppose that there exists an infinite tower

. . . → M i+1 → N i+1 → M i → N i → . . . → N 1 → M of finite covers of M such that for all i ≥ 1, M i → N i is regular, with Galois group H i ' (Z/ 2Z ) r

i

.

If inf i∈N χ h (M i )[π 1 M : π 1 N i ]/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Corollary (6)

Let . . . → M i → M i−1 → . . . → M 1 → M be the tower of finite covers corresponding to the lower mod 2 central series.

For all i ≥ 1, set r i = b 1,F

2

(M i−1 ), with M 0 = M, and R i = r 1 + r 2 + . . . + r i . Suppose that r 1 ≥ 4.

If inf i∈N χ h (M i )2 R

i−1

/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(50)

Virtually infinite first Betti number.

Virtually infinite first Betti number.

Question: Find a tower of finite covers . . . → M i+1 → M i → . . . → M such that lim i→+∞ b 1 (M i ) = +∞ ?

Theorem (5)

Suppose that there exists an infinite tower

. . . → M i+1 → N i+1 → M i → N i → . . . → N 1 → M of finite covers of M such that for all i ≥ 1, M i → N i is regular, with Galois group H i ' (Z/ 2Z ) r

i

.

If inf i∈N χ h (M i )[π 1 M : π 1 N i ]/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Corollary (6)

Let . . . → M i → M i−1 → . . . → M 1 → M be the tower of finite covers corresponding to the lower mod 2 central series.

For all i ≥ 1, set r i = b 1,F

2

(M i−1 ), with M 0 = M, and R i = r 1 + r 2 + . . . + r i . Suppose that r 1 ≥ 4.

If inf i∈N χ h (M i )2 R

i−1

/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(51)

Virtually infinite first Betti number.

Virtually infinite first Betti number.

Question: Find a tower of finite covers . . . → M i+1 → M i → . . . → M such that lim i→+∞ b 1 (M i ) = +∞ ?

Theorem (5)

Suppose that there exists an infinite tower

. . . → M i+1 → N i+1 → M i → N i → . . . → N 1 → M of finite covers of M such that for all i ≥ 1, M i → N i is regular, with Galois group H i ' (Z/ 2Z ) r

i

.

If inf i∈N χ h (M i )[π 1 M : π 1 N i ]/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Corollary (6)

Let . . . → M i → M i−1 → . . . → M 1 → M be the tower of finite covers corresponding to the lower mod 2 central series.

For all i ≥ 1, set r i = b 1,F

2

(M i−1 ), with M 0 = M, and R i = r 1 + r 2 + . . . + r i . Suppose that r 1 ≥ 4.

If inf i∈N χ h (M i )2 R

i−1

/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(52)

Virtually infinite first Betti number.

Virtually infinite first Betti number.

Question: Find a tower of finite covers . . . → M i+1 → M i → . . . → M such that lim i→+∞ b 1 (M i ) = +∞ ?

Theorem (5)

Suppose that there exists an infinite tower

. . . → M i+1 → N i+1 → M i → N i → . . . → N 1 → M of finite covers of M such that for all i ≥ 1, M i → N i is regular, with Galois group H i ' (Z/ 2Z ) r

i

.

If inf i∈N χ h (M i )[π 1 M : π 1 N i ]/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Corollary (6)

Let . . . → M i → M i−1 → . . . → M 1 → M be the tower of finite covers corresponding to the lower mod 2 central series.

For all i ≥ 1, set r i = b 1,F

2

(M i−1 ), with M 0 = M, and R i = r 1 + r 2 + . . . + r i . Suppose that r 1 ≥ 4.

If inf i∈N χ h (M i )2 R

i−1

/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(53)

Virtually infinite first Betti number.

Virtually infinite first Betti number.

Question: Find a tower of finite covers . . . → M i+1 → M i → . . . → M such that lim i→+∞ b 1 (M i ) = +∞ ?

Theorem (5)

Suppose that there exists an infinite tower

. . . → M i+1 → N i+1 → M i → N i → . . . → N 1 → M of finite covers of M such that for all i ≥ 1, M i → N i is regular, with Galois group H i ' (Z/ 2Z ) r

i

.

If inf i∈N χ h (M i )[π 1 M : π 1 N i ]/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Corollary (6)

Let . . . → M i → M i−1 → . . . → M 1 → M be the tower of finite covers corresponding to the lower mod 2 central series.

For all i ≥ 1, set r i = b 1,F

2

(M i−1 ), with M 0 = M, and R i = r 1 + r 2 + . . . + r i . Suppose that r 1 ≥ 4.

If inf i∈N χ h (M i )2 R

i−1

/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(54)

Virtually infinite first Betti number.

Virtually infinite first Betti number.

Question: Find a tower of finite covers . . . → M i+1 → M i → . . . → M such that lim i→+∞ b 1 (M i ) = +∞ ?

Theorem (5)

Suppose that there exists an infinite tower

. . . → M i+1 → N i+1 → M i → N i → . . . → N 1 → M of finite covers of M such that for all i ≥ 1, M i → N i is regular, with Galois group H i ' (Z/ 2Z ) r

i

.

If inf i∈N χ h (M i )[π 1 M : π 1 N i ]/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Corollary (6)

Let . . . → M i → M i−1 → . . . → M 1 → M be the tower of finite covers corresponding to the lower mod 2 central series.

For all i ≥ 1, set r i = b 1,F

2

(M i−1 ), with M 0 = M, and R i = r 1 + r 2 + . . . + r i . Suppose that r 1 ≥ 4.

If inf i∈N χ h (M i )2 R

i−1

/( √ 2) r

i

= 0, Then lim i→+∞ b 1 (M i ) = +∞.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(55)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1

inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2

inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(56)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1

inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2

inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(57)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1

inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2

inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(58)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1

inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2

inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(59)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1

inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2

inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(60)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1

inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2

inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(61)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1

inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2

inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(62)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1 inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2

inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(63)

Virtually infinite first Betti number.

Link with results of Lackenby.

Theorem (Lackenby)

Let . . . → M i → M i−1 → . . . → M 1 → M a tower of finite covers of M such that for all i ≥ 1, M i → M i−1 is regular, of group isomorphic to (Z/ 2Z ) r

i

(with the convention that M 0 = M). Set R i = r 1 + r 2 + . . . + r i .

Suppose that one of the following assumptions is satisfied:

(a) π 1 M does not have property (τ ) with respect to the family {π 1 M i } i∈N (for example if lim i→+∞ h(M i ) = 0) and inf i∈N r

i+1

2

Ri

> 0, or (b) each cover M i → M is regular and lim i→+∞

r

i+1

2

Ri

= +∞.

Then π 1 M is virtually large. In particular, the first Betti number of M is virtually infinite.

Corollary (7)

Let . . . → M i → M i−1 → . . . → M 1 → M be an infinite tower of finite covers of M as in the previous theorem.

Then the first Betti number of M is virtually infinite if neither of the following properties is satisfied.

1 inf i∈N h(M i ) > 0 and the sequence ( r

i+1

2

Ri

) i∈N admits a bounded subsequence.

2 inf i∈N h(M i+1 )4 R

i

( √

2) r

i+1

> 0 and inf i∈N r

i+1

2

Ri

= 0.

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

(64)

Virtually infinite first Betti number.

Thank you for your attention !!!!

Merci pour votre attention !!!!

Claire R Finite covers of a hyperbolic 3-manifold and virtual fibers.

References

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