Hyperbolic volume, Mahler measure, and homology growth

Hyperbolic volume, Mahler measure, and homology growth

Thang Le

School of Mathematics Georgia Institute of Technology

Columbia University, June 2009

Outline

1 Homology Growth and volume

2 Torsion and Determinant

3 L²-Torsion

4 Approximation by finite groups

Outline

1 Homology Growth and volume

2 Torsion and Determinant

3 L²-Torsion

4 Approximation by finite groups

Finite covering of knot complement

K is a knot in S³, X =S³\K , π=π1(X).

π is residually finite: ∃a nested sequence of normal subgroups π=G₀⊃G₁⊃G₂. . .

[π:G_k]<∞, ∩_kG_k ={1}.

If[π:G]<∞, let X_G =G-covering of X X_G^br=branched G-covering of S³ Want: Asymptotics of H₁(X_G^br

k,Z)as k → ∞

.

Finite covering of knot complement

K is a knot in S³, X =S³\K , π=π1(X).

π is residually finite: ∃a nested sequence of normal subgroups π=G₀⊃G₁⊃G₂. . .

[π :G_k]<∞, ∩_kG_k ={1}.

If[π:G]<∞, let X_G =G-covering of X X_G^br=branched G-covering of S³ Want: Asymptotics of H₁(X_G^br

k,Z)as k → ∞

.

Finite covering of knot complement

K is a knot in S³, X =S³\K , π=π1(X).

π is residually finite: ∃a nested sequence of normal subgroups π=G₀⊃G₁⊃G₂. . .

[π :G_k]<∞, ∩_kG_k ={1}.

If[π:G]<∞, let X_G=G-covering of X X_G^br=branched G-covering of S³

Want: Asymptotics of H₁(X_G^br

k,Z)as k → ∞

.

Finite covering of knot complement

K is a knot in S³, X =S³\K , π=π1(X).

π is residually finite: ∃a nested sequence of normal subgroups π=G₀⊃G₁⊃G₂. . .

[π :G_k]<∞, ∩_kG_k ={1}.

If[π:G]<∞, let X_G=G-covering of X X_G^br=branched G-covering of S³ Want: Asymptotics of H₁(X_G^br

k,Z)as k → ∞.

Growth and Volume

(Kazhdan-L ¨uck) lim

k→∞

b₁(X_G^br

k)

[π:G_k] =0 (=L²−Betti number).

t(K,G) :=|TorH₁(X_G^br,Z)|. Definition of Vol(K): X =S³\K is Haken.

X\(ttori) =tpieces each piece is either hyperbolic or Seifert-fibered.

Vol(K) := 1 6π

XVol(hyperbolic pieces) =C(Gromov norm of X). Theorem

lim sup

k→∞

t(K,G_k)^1/[π:Gk]≤exp(Vol(K)).

Growth and Volume

(Kazhdan-L ¨uck) lim

k→∞

b₁(X_G^br

k)

[π:G_k] =0 (=L²−Betti number).

t(K,G) :=|TorH₁(X_G^br,Z)|.

Definition of Vol(K): X =S³\K is Haken. X\(ttori) =tpieces each piece is either hyperbolic or Seifert-fibered.

Vol(K) := 1 6π

XVol(hyperbolic pieces) =C(Gromov norm of X). Theorem

lim sup

k→∞

t(K,G_k)^1/[π:Gk]≤exp(Vol(K)).

Growth and Volume

(Kazhdan-L ¨uck) lim

k→∞

b₁(X_G^br

k)

[π:G_k] =0 (=L²−Betti number).

t(K,G) :=|TorH₁(X_G^br,Z)|.

Definition of Vol(K): X =S³\K is Haken.

X\(ttori) =tpieces each piece is either hyperbolic or Seifert-fibered.

Vol(K) := 1 6π

XVol(hyperbolic pieces) =C(Gromov norm of X).

Theorem

lim sup

k→∞

t(K,G_k)^1/[π:Gk]≤exp(Vol(K)).

Growth and Volume

(Kazhdan-L ¨uck) lim

k→∞

b₁(X_G^br

k)

[π:G_k] =0 (=L²−Betti number).

t(K,G) :=|TorH₁(X_G^br,Z)|.

Definition of Vol(K): X =S³\K is Haken.

X\(ttori) =tpieces each piece is either hyperbolic or Seifert-fibered.

Vol(K) := 1 6π

XVol(hyperbolic pieces) =C(Gromov norm of X).

Theorem

lim sup

k→∞

t(K,G_k)^1/[π:Gk]≤exp(Vol(K)).

Knots with 0 volumes

As a corollary, when Vol(K) =0, we have

k→∞lim t(K,G_k)^1/[π:Gk]=exp(Vol(K)) =1.

Vol(K) =0 if and only if K is in the class i) containing torus knots

ii) closed under connected sum and cabling.

Knots with 0 volumes

As a corollary, when Vol(K) =0, we have

k→∞lim t(K,G_k)^1/[π:Gk]=exp(Vol(K)) =1.

Vol(K) =0 if and only if K is in the class i) containing torus knots

ii) closed under connected sum and cabling.

More general limit: limit as G → ∞

π: a countable group.

S: a finite symmetric set of generators, i.e. g ∈S⇒g⁻¹∈S.

The length of x ∈π:

`_S(x) =smallest length of words representing x S⁰: another symmetric set of generators. Then∃k₁,k₂>0 s.t.

∀x ∈π, k1`<sub>S</sub>(x)< `_S⁰(x)<k2`<sub>S</sub>(x). (`_S and`_S⁰ are quasi-isometric.)

It follows that

n→∞lim `_S(x_n) =∞ ⇐⇒ lim

n→∞`_S⁰(x_n) =∞.

More general limit: limit as G → ∞

π: a countable group.

S: a finite symmetric set of generators, i.e. g ∈S⇒g⁻¹∈S.

The length of x ∈π:

`_S(x) =smallest length of words representing x

S⁰: another symmetric set of generators. Then∃k₁,k₂>0 s.t.

∀x ∈π, k1`<sub>S</sub>(x)< `_S⁰(x)<k2`<sub>S</sub>(x). (`_S and`_S⁰ are quasi-isometric.)

It follows that

n→∞lim `_S(x_n) =∞ ⇐⇒ lim

n→∞`_S⁰(x_n) =∞.

More general limit: limit as G → ∞

π: a countable group.

S: a finite symmetric set of generators, i.e. g ∈S⇒g⁻¹∈S.

The length of x ∈π:

`_S(x) =smallest length of words representing x S⁰: another symmetric set of generators. Then∃k₁,k₂>0 s.t.

∀x ∈π, k₁`<sub>S</sub>(x)< `_S⁰(x)<k₂`_S(x).

(`<sub>S</sub> and`_S⁰ are quasi-isometric.)

It follows that

n→∞lim `_S(x_n) =∞ ⇐⇒ lim

n→∞`_S⁰(x_n) =∞.

More general limit: limit as G → ∞

π: a countable group.

S: a finite symmetric set of generators, i.e. g ∈S⇒g⁻¹∈S.

The length of x ∈π:

`_S(x) =smallest length of words representing x S⁰: another symmetric set of generators. Then∃k₁,k₂>0 s.t.

∀x ∈π, k₁`<sub>S</sub>(x)< `_S⁰(x)<k₂`_S(x).

(`<sub>S</sub> and`_S⁰ are quasi-isometric.) It follows that

n→∞lim `_S(xn) =∞ ⇐⇒ lim

n→∞`_S⁰(xn) =∞.

More general limit

For a subgroup G⊂π, let

diam_S(G) =min{`_S(g),g ∈G\ {1}}.

f : a function defined on a set of finite index normal subgroups ofπ.

diamG→∞lim f(G) =L means there is S such that

lim

diam_SG→∞f(G) =L. Similarly, we can define

lim sup

diamG→∞

f(G).

Remark: If lim_k→∞diamG=∞then∩G_k ={1} (co-final).

More general limit

For a subgroup G⊂π, let

diam_S(G) =min{`_S(g),g ∈G\ {1}}.

f : a function defined on a set of finite index normal subgroups ofπ.

diamG→∞lim f(G) =L means there is S such that

lim

diam_SG→∞f(G) =L.

Similarly, we can define

lim sup

diamG→∞

f(G).

Remark: If lim_k→∞diamG=∞then∩G_k ={1} (co-final).

More general limit

For a subgroup G⊂π, let

diam_S(G) =min{`_S(g),g ∈G\ {1}}.

f : a function defined on a set of finite index normal subgroups ofπ.

diamG→∞lim f(G) =L means there is S such that

lim

diam_SG→∞f(G) =L.

Similarly, we can define

lim sup

diamG→∞

f(G).

Remark: If lim_k→∞diamG=∞then∩G_k ={1} (co-final).

More general limit

For a subgroup G⊂π, let

diam_S(G) =min{`_S(g),g ∈G\ {1}}.

f : a function defined on a set of finite index normal subgroups ofπ.

diamG→∞lim f(G) =L means there is S such that

lim

diam_SG→∞f(G) =L.

Similarly, we can define

lim sup

diamG→∞

f(G).

Remark: If lim_k→∞diamG=∞then∩G_k ={1} (co-final).

Homology Growth and Volume

Conjecture

(“volume conjecture”) For every knot K ⊂S³, lim sup

G→∞

t(K,G)^1/[π:G]=exp(Vol(K)).

True: LHS≤RHS. True for knots with Vol=0.

To prove the conjecture one needs to find{G_k}– finite index normal subgroups ofπ s. t. lim_kdiam(G_k) =∞and

lim

k→∞t(K,G_k)^1/[π:Gk]=exp(Vol(K)). (*) It is unlikely that for any sequence G_k of normal subgroups s.t.

lim diamG_k =∞one has (*). Which{G_k}should we choose?

Homology Growth and Volume

Conjecture

(“volume conjecture”) For every knot K ⊂S³, lim sup

G→∞

t(K,G)^1/[π:G]=exp(Vol(K)).

True: LHS≤RHS. True for knots with Vol=0.

To prove the conjecture one needs to find{G_k}– finite index normal subgroups ofπ s. t. lim_kdiam(G_k) =∞and

lim

k→∞t(K,G_k)^1/[π:Gk]=exp(Vol(K)). (*) It is unlikely that for any sequence G_k of normal subgroups s.t.

lim diamG_k =∞one has (*). Which{G_k}should we choose?

Homology Growth and Volume

Conjecture

(“volume conjecture”) For every knot K ⊂S³, lim sup

G→∞

t(K,G)^1/[π:G]=exp(Vol(K)).

True: LHS≤RHS. True for knots with Vol=0.

To prove the conjecture one needs to find{G_k}– finite index normal subgroups ofπ s. t. lim_kdiam(G_k) =∞and

lim

k→∞t(K,G_k)^1/[π:Gk]=exp(Vol(K)). (*)

It is unlikely that for any sequence G_k of normal subgroups s.t. lim diamG_k =∞one has (*). Which{G_k}should we choose?

Homology Growth and Volume

Conjecture

(“volume conjecture”) For every knot K ⊂S³, lim sup

G→∞

t(K,G)^1/[π:G]=exp(Vol(K)).

True: LHS≤RHS. True for knots with Vol=0.

To prove the conjecture one needs to find{G_k}– finite index normal subgroups ofπ s. t. lim_kdiam(G_k) =∞and

lim

k→∞t(K,G_k)^1/[π:Gk]=exp(Vol(K)). (*) It is unlikely that for any sequence G_k of normal subgroups s.t.

lim diamG_k =∞one has (*). Which{G_k}should we choose?

Expander family

Long-Lubotzky-Reid (2007): ∀hyperbolic knot,∃ {G_k}– finite index normal subgroups, such that

π has propertyτ w.r.t.{G_k}.

⇔Cayley graphs ofπ/G_k w.r.t. a fixed symmetric set of generators form a family of expanders

⇔the least non-zero eigenvalue of the Laplacian of the Cayley graphs ofπ/G_k is≥a fixed >0.

Based on deep results of Bourgain-Gamburg (2007) on expanders from SL(2,p).

Conjecture

(*) holds for the Long-Lubotzky-Reid sequence{G_k}. Justification: to follow.

Expander family

Long-Lubotzky-Reid (2007): ∀hyperbolic knot,∃ {G_k}– finite index normal subgroups, such that

π has propertyτ w.r.t.{G_k}.

⇔Cayley graphs ofπ/G_k w.r.t. a fixed symmetric set of generators form a family of expanders

⇔the least non-zero eigenvalue of the Laplacian of the Cayley graphs ofπ/G_k is≥a fixed >0.

Based on deep results of Bourgain-Gamburg (2007) on expanders from SL(2,p).

Conjecture

(*) holds for the Long-Lubotzky-Reid sequence{G_k}. Justification: to follow.

Expander family

Long-Lubotzky-Reid (2007): ∀hyperbolic knot,∃ {G_k}– finite index normal subgroups, such that

π has propertyτ w.r.t.{G_k}.

⇔Cayley graphs ofπ/G_k w.r.t. a fixed symmetric set of generators form a family of expanders

⇔the least non-zero eigenvalue of the Laplacian of the Cayley graphs ofπ/G_k is≥a fixed >0.

Based on deep results of Bourgain-Gamburg (2007) on expanders from SL(2,p).

Conjecture

(*) holds for the Long-Lubotzky-Reid sequence{G_k}. Justification: to follow.

Expander family

Long-Lubotzky-Reid (2007): ∀hyperbolic knot,∃ {G_k}– finite index normal subgroups, such that

π has propertyτ w.r.t.{G_k}.

⇔Cayley graphs ofπ/G_k w.r.t. a fixed symmetric set of generators form a family of expanders

⇔the least non-zero eigenvalue of the Laplacian of the Cayley graphs ofπ/G_k is≥a fixed >0.

Based on deep results of Bourgain-Gamburg (2007) on expanders from SL(2,p).

Conjecture

(*) holds for the Long-Lubotzky-Reid sequence{G_k}. Justification: to follow.

Expander family

Long-Lubotzky-Reid (2007): ∀hyperbolic knot,∃ {G_k}– finite index normal subgroups, such that

π has propertyτ w.r.t.{G_k}.

⇔Cayley graphs ofπ/G_k w.r.t. a fixed symmetric set of generators form a family of expanders

⇔the least non-zero eigenvalue of the Laplacian of the Cayley graphs ofπ/G_k is≥a fixed >0.

Based on deep results of Bourgain-Gamburg (2007) on expanders from SL(2,p).

Conjecture

(*) holds for the Long-Lubotzky-Reid sequence{G_k}.

Justification: to follow.

Outline

1 Homology Growth and volume

2 Torsion and Determinant

3 L²-Torsion

4 Approximation by finite groups

Reidemeister Torsion

C: Chain complex of finite dimensionalC-modules (vector spaces).

0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

SupposeCisacyclicandbased. Then the torsionτ(C)is defined.

c_i : base of C_i. Each∂i is given by a matrix. Simplest case: Cis 0→C₁→^∂1 C₀→0.

τ(C) =det∂₁. 0→C₂→^∂2 C₁→^∂1 C₀→0.

τ(C) =

∂₂(c₂)∂⁻¹c₀ c₁

Here[a/b]is the determinant of the change matrix from b to a.

Reidemeister Torsion

C: Chain complex of finite dimensionalC-modules (vector spaces).

0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

SupposeCisacyclicandbased. Then the torsionτ(C)is defined.

c_i : base of C_i. Each∂_i is given by a matrix.

Simplest case: Cis 0→C₁→^∂1 C₀→0. τ(C) =det∂₁. 0→C₂→^∂2 C₁→^∂1 C₀→0.

τ(C) =

∂₂(c₂)∂⁻¹c₀ c₁

Here[a/b]is the determinant of the change matrix from b to a.

Reidemeister Torsion

C: Chain complex of finite dimensionalC-modules (vector spaces).

0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

SupposeCisacyclicandbased. Then the torsionτ(C)is defined.

c_i : base of C_i. Each∂_i is given by a matrix.

Simplest case: Cis 0→C₁→^∂1 C₀→0.

τ(C) =det∂₁.

0→C₂→^∂2 C₁→^∂1 C₀→0. τ(C) =

∂₂(c₂)∂⁻¹c₀ c₁

Here[a/b]is the determinant of the change matrix from b to a.

Reidemeister Torsion

C: Chain complex of finite dimensionalC-modules (vector spaces).

0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

SupposeCisacyclicandbased. Then the torsionτ(C)is defined.

c_i : base of C_i. Each∂_i is given by a matrix.

Simplest case: Cis 0→C₁→^∂1 C₀→0.

τ(C) =det∂₁. 0→C₂→^∂2 C₁→^∂1 C₀→0.

τ(C) =

∂₂(c₂)∂⁻¹c₀ c₁

Here[a/b]is the determinant of the change matrix from b to a.

Torsion of chain of Hilbert spaces

C: complex of finite dimensional Hilbert spaces overC;acyclic.

Choose orthonormal base c_i for each C_i, defineτ(C,c).

Change of base: τ(C) :=|τ(C,c)|is well-defined.

C: complex of Hilbert spaces overC[π]. Want to defineτ(C). 0→Cn ∂n

→Cn−1

∂n−1

→ . . .C1

∂1

→C0→0. More specifically,

C_i =Z[π]ⁿⁱ, freeZ[π]−module, or C_i =`²(π)ⁿⁱ

∂i ∈Mat(n_i×n_i−1,Z[π]), acting on the right. Need to define what is the determinant of a matrix A∈Mat(m×n,Z[π]).

Torsion of chain of Hilbert spaces

C: complex of finite dimensional Hilbert spaces overC;acyclic.

Choose orthonormal base c_i for each C_i, defineτ(C,c).

Change of base: τ(C) :=|τ(C,c)|is well-defined.

C: complex of Hilbert spaces overC[π]. Want to defineτ(C).

0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

More specifically,

C_i =Z[π]ⁿⁱ, freeZ[π]−module, or C_i =`²(π)ⁿⁱ

∂i ∈Mat(n_i×n_i−1,Z[π]), acting on the right.

Need to define what is the determinant of a matrix A∈Mat(m×n,Z[π]).

Torsion of chain of Hilbert spaces

C: complex of finite dimensional Hilbert spaces overC;acyclic.

Choose orthonormal base c_i for each C_i, defineτ(C,c).

Change of base: τ(C) :=|τ(C,c)|is well-defined.

C: complex of Hilbert spaces overC[π]. Want to defineτ(C).

0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

More specifically,

C_i =Z[π]ⁿⁱ, freeZ[π]−module, or C_i =`²(π)ⁿⁱ

∂i ∈Mat(n_i×n_i−1,Z[π]), acting on the right.

Need to define what is the determinant of a matrix A∈Mat(m×n,Z[π]).

Trace on C [π]

For square matrix A with complex entries: log det A=tr log A.

One can define a good theory of determinant of there is a good trace.

Regular representation: C[π]acts on the right on the Hilbert space

`²(π) ={X

g

cgg |X

|c_g|²<∞}.

Remark. Ifπ=π1(S³\K), K is not a torus knot, then the regular representation is of type II₁.

Adjoint operator: x =P

cgg∈C[π], then x^∗ =Pc¯gg⁻¹. Similarly to the finite group case, define∀g ∈π,

tr(g) =δ_g,1

∀x ∈C[π],tr(x) =hx,1i=coeff. of 1 in x.

Trace on C [π]

For square matrix A with complex entries: log det A=tr log A.

One can define a good theory of determinant of there is a good trace.

Regular representation: C[π]acts on the right on the Hilbert space

`²(π) ={X

g

cgg |X

|c_g|²<∞}.

Remark. Ifπ=π1(S³\K), K is not a torus knot, then the regular representation is of type II₁.

Adjoint operator: x =P

cgg∈C[π], then x^∗ =Pc¯gg⁻¹. Similarly to the finite group case, define∀g ∈π,

tr(g) =δ_g,1

∀x ∈C[π],tr(x) =hx,1i=coeff. of 1 in x.

Trace on C [π]

For square matrix A with complex entries: log det A=tr log A.

One can define a good theory of determinant of there is a good trace.

Regular representation: C[π]acts on the right on the Hilbert space

`²(π) ={X

g

cgg |X

|c_g|²<∞}.

Remark. Ifπ=π1(S³\K), K is not a torus knot, then the regular representation is of type II₁.

Adjoint operator: x =P

cgg∈C[π], then x^∗ =Pc¯gg⁻¹. Similarly to the finite group case, define∀g ∈π,

tr(g) =δ_g,1

∀x ∈C[π],tr(x) =hx,1i=coeff. of 1 in x.

Trace on C [π]

For square matrix A with complex entries: log det A=tr log A.

One can define a good theory of determinant of there is a good trace.

Regular representation: C[π]acts on the right on the Hilbert space

`²(π) ={X

g

cgg |X

|c_g|²<∞}.

Remark. Ifπ=π1(S³\K), K is not a torus knot, then the regular representation is of type II₁.

Adjoint operator: x =P

cgg∈C[π], then x^∗ =Pc¯gg⁻¹.

Similarly to the finite group case, define∀g ∈π, tr(g) =δ_g,1

∀x ∈C[π],tr(x) =hx,1i=coeff. of 1 in x.

Trace on C [π]

For square matrix A with complex entries: log det A=tr log A.

One can define a good theory of determinant of there is a good trace.

Regular representation: C[π]acts on the right on the Hilbert space

`²(π) ={X

g

cgg |X

|c_g|²<∞}.

Remark. Ifπ=π1(S³\K), K is not a torus knot, then the regular representation is of type II₁.

Adjoint operator: x =P

cgg∈C[π], then x^∗ =Pc¯gg⁻¹. Similarly to the finite group case, define∀g ∈π,

tr(g) =δ_g,1

∀x ∈C[π],tr(x) =hx,1i=coeff. of 1 in x.

Trace

The trace can be extended to the Von Neumann algebraN(π)⊃C[π].

A∈Mat(n×n,C[π]). Define tr(A) :=

n

X

i=1

tr(A_ii). (not rigorous) Define det(A)using

log det A=tr log A

=−tr

∞

X

p=1

(I−A)^p/p

=−Xtr[(I−A)^p]

p .

Convergence of the RHS?

Trace

The trace can be extended to the Von Neumann algebraN(π)⊃C[π].

A∈Mat(n×n,C[π]). Define tr(A) :=

n

X

i=1

tr(A_ii).

(not rigorous) Define det(A)using log det A=tr log A

=−tr

∞

X

p=1

(I−A)^p/p

=−Xtr[(I−A)^p]

p .

Convergence of the RHS?

Trace

The trace can be extended to the Von Neumann algebraN(π)⊃C[π].

A∈Mat(n×n,C[π]). Define tr(A) :=

n

X

i=1

tr(A_ii).

(not rigorous) Define det(A)using log det A=tr log A

=−tr

∞

X

p=1

(I−A)^p/p

=−Xtr[(I−A)^p]

p .

Convergence of the RHS?

Trace

The trace can be extended to the Von Neumann algebraN(π)⊃C[π].

A∈Mat(n×n,C[π]). Define tr(A) :=

n

X

i=1

tr(A_ii).

(not rigorous) Define det(A)using log det A=tr log A

=−tr

∞

X

p=1

(I−A)^p/p

=−Xtr[(I−A)^p]

p .

Convergence of the RHS?

Fuglede-Kadison-L ¨uck determinant for A ∈ Mat(m × n, C [π])

B:=A^∗A, where(A^∗)_ij := (A_ji)^∗. ker(B) =ker A, B ≥0.

Choose k >||B||. Let C=B/k . I≥I−C≥0, and (I−C)^p ≥(I−C)^p+1≥0.

The sequence tr[(I−C)^p]is decreasing⇒lim tr[(I−C)^p] =b≥0. b=b(A)depends only on A, equal to the Von- Neumann

dimension of ker A.

Use b as the correction term in the log series to define det_πC: log det_πC=−X1

p(tr[(I−C)^p]−b) =finite or− ∞. B=kC, det_πB=k^n−bdet C ∈R≥0, det_πA=√

det_πB. Most interesting case: A is injective (b =0), m=n, but not invertible.

Fuglede-Kadison-L ¨uck determinant for A ∈ Mat(m × n, C [π])

B:=A^∗A, where(A^∗)_ij := (A_ji)^∗. ker(B) =ker A, B ≥0.

Choose k >||B||. Let C =B/k . I≥I−C≥0, and (I−C)^p≥(I−C)^p+1≥0.

The sequence tr[(I−C)^p]is decreasing⇒lim tr[(I−C)^p] =b≥0. b=b(A)depends only on A, equal to the Von- Neumann

dimension of ker A.

Use b as the correction term in the log series to define det_πC: log det_πC=−X1

p(tr[(I−C)^p]−b) =finite or− ∞. B=kC, det_πB=k^n−bdet C ∈R≥0, det_πA=√

det_πB. Most interesting case: A is injective (b =0), m=n, but not invertible.

Fuglede-Kadison-L ¨uck determinant for A ∈ Mat(m × n, C [π])

B:=A^∗A, where(A^∗)_ij := (A_ji)^∗. ker(B) =ker A, B ≥0.

Choose k >||B||. Let C =B/k . I≥I−C≥0, and (I−C)^p≥(I−C)^p+1≥0.

The sequence tr[(I−C)^p]is decreasing⇒lim tr[(I−C)^p] =b≥0.

b=b(A)depends only on A, equal to the Von- Neumann dimension of ker A.

Use b as the correction term in the log series to define det_πC: log det_πC=−X1

p(tr[(I−C)^p]−b) =finite or− ∞. B=kC, det_πB=k^n−bdet C ∈R≥0, det_πA=√

det_πB. Most interesting case: A is injective (b =0), m=n, but not invertible.

Fuglede-Kadison-L ¨uck determinant for A ∈ Mat(m × n, C [π])

B:=A^∗A, where(A^∗)_ij := (A_ji)^∗. ker(B) =ker A, B ≥0.

Choose k >||B||. Let C =B/k . I≥I−C≥0, and (I−C)^p≥(I−C)^p+1≥0.

The sequence tr[(I−C)^p]is decreasing⇒lim tr[(I−C)^p] =b≥0.

b=b(A)depends only on A, equal to the Von- Neumann dimension of ker A.

Use b as the correction term in the log series to define det_πC:

log det_πC=−X1

p(tr[(I−C)^p]−b) =finite or− ∞.

B=kC, det_πB=k^n−bdet C ∈R≥0, det_πA=√ det_πB. Most interesting case: A is injective (b =0), m=n, but not invertible.

Fuglede-Kadison-L ¨uck determinant for A ∈ Mat(m × n, C [π])

B:=A^∗A, where(A^∗)_ij := (A_ji)^∗. ker(B) =ker A, B ≥0.

Choose k >||B||. Let C =B/k . I≥I−C≥0, and (I−C)^p≥(I−C)^p+1≥0.

The sequence tr[(I−C)^p]is decreasing⇒lim tr[(I−C)^p] =b≥0.

b=b(A)depends only on A, equal to the Von- Neumann dimension of ker A.

Use b as the correction term in the log series to define det_πC:

log det_πC=−X1

p(tr[(I−C)^p]−b) =finite or− ∞.

B=kC, det_πB=k^n−bdet C ∈R≥0, det_πA=√ det_πB.

Most interesting case: A is injective (b =0), m=n, but not invertible.

Fuglede-Kadison-L ¨uck determinant for A ∈ Mat(m × n, C [π])

B:=A^∗A, where(A^∗)_ij := (A_ji)^∗. ker(B) =ker A, B ≥0.

Choose k >||B||. Let C =B/k . I≥I−C≥0, and (I−C)^p≥(I−C)^p+1≥0.

The sequence tr[(I−C)^p]is decreasing⇒lim tr[(I−C)^p] =b≥0.

b=b(A)depends only on A, equal to the Von- Neumann dimension of ker A.

Use b as the correction term in the log series to define det_πC:

log det_πC=−X1

p(tr[(I−C)^p]−b) =finite or− ∞.

B=kC, det_πB=k^n−bdet C ∈R≥0, det_πA=√ det_πB.

Most interesting case: A is injective (b =0), m=n, but not invertible.

FKL determinant – Example: Finite group

D∈Mat(n×n,C). Let p(λ) =det(λI+D).

det⁰D:=coeff. of smallest degree of p= Y

λeigenvalue6=0

λ.

π ={1}, A∈Mat(m×n,C). Then in general det_{1}A6=det A. det_{1}A=

q

det⁰(A^∗A) =Y

(non-zero singular values).

|π|<∞, A∈Mat(m×n,C[π]). Then A is given by a matrix D∈Mat(m|π| ×n|π|,C).

det_πA= det⁰(D^∗D)1/2|π|

.

FKL determinant – Example: Finite group

D∈Mat(n×n,C). Let p(λ) =det(λI+D).

det⁰D:=coeff. of smallest degree of p= Y

λeigenvalue6=0

λ.

π ={1}, A∈Mat(m×n,C). Then in general det_{1}A6=det A.

det_{1}A= q

det⁰(A^∗A) =Y

(non-zero singular values).

|π|<∞, A∈Mat(m×n,C[π]). Then A is given by a matrix D∈Mat(m|π| ×n|π|,C).

det_πA= det⁰(D^∗D)1/2|π|

.

FKL determinant – Example: Finite group

D∈Mat(n×n,C). Let p(λ) =det(λI+D).

det⁰D:=coeff. of smallest degree of p= Y

λeigenvalue6=0

λ.

π ={1}, A∈Mat(m×n,C). Then in general det_{1}A6=det A.

det_{1}A= q

det⁰(A^∗A) =Y

(non-zero singular values).

|π|<∞, A∈Mat(m×n,C[π]). Then A is given by a matrix D∈Mat(m|π| ×n|π|,C).

det_πA= det⁰(D^∗D)1/2|π|

.

FKL determinant– Example: π = Z

f(t₁^±1, . . . ,t_µ^±1)∈C[Z^µ]≡C[t₁^±1, . . . ,t_µ^±1].

Assume f 6=0. f : 1×1 matrix.

It is known that (L ¨uck) det_Z^µ(f)is the Mahler measure: det_Z^µf =M(f) :=exp

Z

T^µ

log|f|dσ

whereT^µ={(z₁, . . . ,z_µ)∈C^µ| |z_i|=1}, theµ-torus. dσ: the invariant measure normalized so thatR

T^µdσ =1. f(t)∈Z[t^±1], f(t) =a₀Qn

j=1(t−z_j),z_j ∈C. Then M(f) =a₀ Y

|z_j|>1

|z_j|.

FKL determinant– Example: π = Z

f(t₁^±1, . . . ,t_µ^±1)∈C[Z^µ]≡C[t₁^±1, . . . ,t_µ^±1].

Assume f 6=0. f : 1×1 matrix.

It is known that (L ¨uck) det_Z^µ(f)is the Mahler measure:

det_Z^µf =M(f) :=exp Z

T^µ

log|f|dσ

whereT^µ={(z₁, . . . ,z_µ)∈C^µ| |z_i|=1}, theµ-torus.

dσ: the invariant measure normalized so thatR

T^µdσ=1.

f(t)∈Z[t^±1], f(t) =a₀Qn

j=1(t−z_j),z_j ∈C. Then M(f) =a₀ Y

|z_j|>1

|z_j|.

FKL determinant– Example: π = Z

f(t₁^±1, . . . ,t_µ^±1)∈C[Z^µ]≡C[t₁^±1, . . . ,t_µ^±1].

Assume f 6=0. f : 1×1 matrix.

It is known that (L ¨uck) det_Z^µ(f)is the Mahler measure:

det_Z^µf =M(f) :=exp Z

T^µ

log|f|dσ

whereT^µ={(z₁, . . . ,z_µ)∈C^µ| |z_i|=1}, theµ-torus.

dσ: the invariant measure normalized so thatR

T^µdσ=1.

f(t)∈Z[t^±1], f(t) =a₀Qn

j=1(t−z_j),z_j ∈C. Then M(f) =a₀ Y

|z_j|>1

|z_j|.

Outline

1 Homology Growth and volume

2 Torsion and Determinant

3 L²-Torsion

4 Approximation by finite groups

L

-Torsion, L

-homology of C [π]- complex

C: 0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

C_i =`²(π)ⁿⁱ, ∂_i ∈Mat(n_i×ni−1,C[π]).

Cis ofdet-classif det_π(∂i)6=0∀i. In that case τ⁽²⁾(C) := det_π(∂₁) det_π(∂₃) det_π(∂₅). . .

det_π(∂₂) det_π(∂₄). . . . L²-homology (no need to be of det-class)

H_i⁽²⁾:=ker∂_i/Im(∂_i−1). Cis L²-acyclic if H_i⁽²⁾=0∀i.

L

-Torsion, L

-homology of C [π]- complex

C: 0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

C_i =`²(π)ⁿⁱ, ∂_i ∈Mat(n_i×ni−1,C[π]).

Cis ofdet-classif det_π(∂i)6=0∀i. In that case τ⁽²⁾(C) := det_π(∂₁) det_π(∂₃) det_π(∂₅). . .

det_π(∂2) det_π(∂4). . . .

L²-homology (no need to be of det-class) H_i⁽²⁾:=ker∂_i/Im(∂_i−1). Cis L²-acyclic if H_i⁽²⁾=0∀i.

L

-Torsion, L

-homology of C [π]- complex

C: 0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

C_i =`²(π)ⁿⁱ, ∂_i ∈Mat(n_i×ni−1,C[π]).

Cis ofdet-classif det_π(∂i)6=0∀i. In that case τ⁽²⁾(C) := det_π(∂₁) det_π(∂₃) det_π(∂₅). . .

det_π(∂2) det_π(∂4). . . . L²-homology (no need to be of det-class)

H_i⁽²⁾:=ker∂_i/Im(∂_i−1).

Cis L²-acyclic if H_i⁽²⁾=0∀i.

L

-Torsion, L

-homology of C [π]- complex

C: 0→Cn ∂n

→Cn−1

∂n−1

→ . . .C₁→^∂1 C₀→0.

C_i =`²(π)ⁿⁱ, ∂_i ∈Mat(n_i×ni−1,C[π]).

Cis ofdet-classif det_π(∂i)6=0∀i. In that case τ⁽²⁾(C) := det_π(∂₁) det_π(∂₃) det_π(∂₅). . .

det_π(∂2) det_π(∂4). . . . L²-homology (no need to be of det-class)

H_i⁽²⁾:=ker∂_i/Im(∂_i−1).

Cis L²-acyclic if H_i⁽²⁾=0∀i.

L

-Torsion of manifolds: Definition

X is a˜ π-space such that p: ˜X →X := ˜X/πis a regular covering.

X˜,X manifold.

Finite triangulation of X : C( ˜X)becomes a complex of free Z[π]-modules.

If C( ˜X)is of det-class, then L²-torsion, denoted byτ⁽²⁾( ˜X), can be defined. Depends on the triangulation.

If C( ˜X)isacyclicand ofdet-classfor one triangulation, then it is acyclic and of det-class for any other triangulation, andτ⁽²⁾( ˜X)of the two triangulations are the same: we can defineτ⁽²⁾( ˜X).

L

-Torsion of manifolds: Definition

X is a˜ π-space such that p: ˜X →X := ˜X/πis a regular covering.

X˜,X manifold.

Finite triangulation of X : C( ˜X)becomes a complex of free Z[π]-modules.

If C( ˜X)is of det-class, then L²-torsion, denoted byτ⁽²⁾( ˜X), can be defined. Depends on the triangulation.

If C( ˜X)isacyclicand ofdet-classfor one triangulation, then it is acyclic and of det-class for any other triangulation, andτ⁽²⁾( ˜X)of the two triangulations are the same: we can defineτ⁽²⁾( ˜X).

L

-Torsion of manifolds: Definition

X is a˜ π-space such that p: ˜X →X := ˜X/πis a regular covering.

X˜,X manifold.

Finite triangulation of X : C( ˜X)becomes a complex of free Z[π]-modules.

If C( ˜X)is of det-class, then L²-torsion, denoted byτ⁽²⁾( ˜X), can be defined. Depends on the triangulation.

If C( ˜X)isacyclicand ofdet-classfor one triangulation, then it is acyclic and of det-class for any other triangulation, andτ⁽²⁾( ˜X)of the two triangulations are the same: we can defineτ⁽²⁾( ˜X).

L

-Torsion of knots: universal covering

K a knot in S³. X =S³−K, X : universal covering.˜ π =π₁(X). ThenX is a˜ π-space with quotient X .

C( ˜X)is acyclic and is of det-class.

τ⁽²⁾(K) :=τ⁽²⁾( ˜X). Theorem(L ¨uck-Schick)

logτ⁽²⁾(K) =−Vol(K).

based on results of Burghelea-Friedlander-Kappeler-McDonald, Lott, and Mathai.

L

-Torsion of knots: universal covering

K a knot in S³. X =S³−K, X : universal covering.˜ π =π₁(X). ThenX is a˜ π-space with quotient X .

C( ˜X)is acyclic and is of det-class.

τ⁽²⁾(K) :=τ⁽²⁾( ˜X).

Theorem(L ¨uck-Schick)

logτ⁽²⁾(K) =−Vol(K).

based on results of Burghelea-Friedlander-Kappeler-McDonald, Lott, and Mathai.

L

-Torsion of knots: universal covering

K a knot in S³. X =S³−K, X : universal covering.˜ π =π₁(X). ThenX is a˜ π-space with quotient X .

C( ˜X)is acyclic and is of det-class.

τ⁽²⁾(K) :=τ⁽²⁾( ˜X).

Theorem(L ¨uck-Schick)

logτ⁽²⁾(K) =−Vol(K).

based on results of Burghelea-Friedlander-Kappeler-McDonald, Lott, and Mathai.

L

-Torsion of knots: computing using knot group

π =π₁(S³\K).

π=ha₁, . . . ,a_n+1|r₁, . . . ,r_ni.

Y : 2-CW complex associated with this presentation. X and Y are homotopic.

Y has 1 0-cell,(n+1)1-cells, and n 2-cells.Y : universal covering.˜ C( ˜Y) : 0→Z[π]ⁿ−→^∂2 Z[π]ⁿ⁺¹−→^∂1 Z[π]→0.

∂₁=











a₁−1 a₂−1

... a_n+1−1











, ∂₂= ∂r_i

∂a_j

∈Mat(n×(n+1),Z[π])

L

-Torsion of knots: computing using knot group

π =π₁(S³\K).

π=ha₁, . . . ,a_n+1|r₁, . . . ,r_ni.

Y : 2-CW complex associated with this presentation. X and Y are homotopic.

Y has 1 0-cell,(n+1)1-cells, and n 2-cells.Y : universal covering.˜ C( ˜Y) : 0→Z[π]ⁿ−→^∂2 Z[π]ⁿ⁺¹−→^∂1 Z[π]→0.

∂₁=











a₁−1 a₂−1

... a_n+1−1











, ∂₂= ∂r_i

∂a_j

∈Mat(n×(n+1),Z[π])