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On l 2 -Betti numbers and their analogues in positive characteristic - - PowerPoint PPT Presentation

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On l 2 -Betti numbers and their analogues in positive characteristic Andrei Jaikin-Zapirain Birmingham, August 12th, 2017 L 2 -Betti numbers Andrei Jaikin-Zapirain The initial setting G is a finitely generated group. G > G 1 > G 2 > .

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SLIDE 1

On l2-Betti numbers and their analogues in positive characteristic

Andrei Jaikin-Zapirain Birmingham, August 12th, 2017

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 2

The initial setting

G is a finitely generated group. G > G1 > G2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Matn×m(K[G]) φA

G/Gi :

K[G/Gi]n → K[G/Gi]m (v1, . . . , vn) → (v1, . . . , vn)A. rkG/Gi(A) =

dimK Im φA

G/Gi

|G:Gi|

= n −

dimK ker φA

G/Gi

|G:Gi|

{rkG/Gi} is a collection of Sylvester matrix rank functions on K[G].

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 3

The initial setting

G is a finitely generated group. G > G1 > G2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Matn×m(K[G]) φA

G/Gi :

K[G/Gi]n → K[G/Gi]m (v1, . . . , vn) → (v1, . . . , vn)A. rkG/Gi(A) =

dimK Im φA

G/Gi

|G:Gi|

= n −

dimK ker φA

G/Gi

|G:Gi|

{rkG/Gi} is a collection of Sylvester matrix rank functions on K[G].

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 4

The initial setting

G is a finitely generated group. G > G1 > G2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Matn×m(K[G]) φA

G/Gi :

K[G/Gi]n → K[G/Gi]m (v1, . . . , vn) → (v1, . . . , vn)A. rkG/Gi(A) =

dimK Im φA

G/Gi

|G:Gi|

= n −

dimK ker φA

G/Gi

|G:Gi|

{rkG/Gi} is a collection of Sylvester matrix rank functions on K[G].

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 5

The initial setting

G is a finitely generated group. G > G1 > G2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Matn×m(K[G]) φA

G/Gi :

K[G/Gi]n → K[G/Gi]m (v1, . . . , vn) → (v1, . . . , vn)A. rkG/Gi(A) =

dimK Im φA

G/Gi

|G:Gi|

= n −

dimK ker φA

G/Gi

|G:Gi|

{rkG/Gi} is a collection of Sylvester matrix rank functions on K[G].

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 6

The initial setting

G is a finitely generated group. G > G1 > G2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Matn×m(K[G]) φA

G/Gi :

K[G/Gi]n → K[G/Gi]m (v1, . . . , vn) → (v1, . . . , vn)A. rkG/Gi(A) =

dimK Im φA

G/Gi

|G:Gi|

= n −

dimK ker φA

G/Gi

|G:Gi|

{rkG/Gi} is a collection of Sylvester matrix rank functions on K[G].

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 7

The initial setting

G is a finitely generated group. G > G1 > G2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Matn×m(K[G]) φA

G/Gi :

K[G/Gi]n → K[G/Gi]m (v1, . . . , vn) → (v1, . . . , vn)A. rkG/Gi(A) =

dimK Im φA

G/Gi

|G:Gi|

= n −

dimK ker φA

G/Gi

|G:Gi|

{rkG/Gi} is a collection of Sylvester matrix rank functions on K[G].

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 8

The initial setting

G is a finitely generated group. G > G1 > G2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Matn×m(K[G]) φA

G/Gi :

K[G/Gi]n → K[G/Gi]m (v1, . . . , vn) → (v1, . . . , vn)A. rkG/Gi(A) =

dimK Im φA

G/Gi

|G:Gi|

= n −

dimK ker φA

G/Gi

|G:Gi|

{rkG/Gi} is a collection of Sylvester matrix rank functions on K[G].

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 9

Sylvester rank function on a K-algebra

Let R be a K-algebra. A Sylvester matrix rank function rk on R is a map rk : Mat(R) → R≥0 satisfying the following conditions (SRF1) rk(M) = 0 if M is any zero matrix and rk(1R) = 1; (SRF2) rk(M1M2) ≤ min{rk(M1), rk(M2)} if M1 and M2 can be multiplied; (SRF3) rk M1 M2

  • = rk(M1) + rk(M2);

(SRF4) rk M1 M3 M2

  • ≥ rk(M1) + rk(M2) if M1, M2 and

M3 are of appropriate sizes. The space P(R) of Sylvester rank functions on R is a compact convex subset of RMat(R)

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 10

Sylvester rank function on a K-algebra

Let R be a K-algebra. A Sylvester matrix rank function rk on R is a map rk : Mat(R) → R≥0 satisfying the following conditions (SRF1) rk(M) = 0 if M is any zero matrix and rk(1R) = 1; (SRF2) rk(M1M2) ≤ min{rk(M1), rk(M2)} if M1 and M2 can be multiplied; (SRF3) rk M1 M2

  • = rk(M1) + rk(M2);

(SRF4) rk M1 M3 M2

  • ≥ rk(M1) + rk(M2) if M1, M2 and

M3 are of appropriate sizes. The space P(R) of Sylvester rank functions on R is a compact convex subset of RMat(R)

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 11

Sylvester rank function on a K-algebra

Let R be a K-algebra. A Sylvester matrix rank function rk on R is a map rk : Mat(R) → R≥0 satisfying the following conditions (SRF1) rk(M) = 0 if M is any zero matrix and rk(1R) = 1; (SRF2) rk(M1M2) ≤ min{rk(M1), rk(M2)} if M1 and M2 can be multiplied; (SRF3) rk M1 M2

  • = rk(M1) + rk(M2);

(SRF4) rk M1 M3 M2

  • ≥ rk(M1) + rk(M2) if M1, M2 and

M3 are of appropriate sizes. The space P(R) of Sylvester rank functions on R is a compact convex subset of RMat(R)

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 12

Sylvester rank function on a K-algebra

Let R be a K-algebra. A Sylvester matrix rank function rk on R is a map rk : Mat(R) → R≥0 satisfying the following conditions (SRF1) rk(M) = 0 if M is any zero matrix and rk(1R) = 1; (SRF2) rk(M1M2) ≤ min{rk(M1), rk(M2)} if M1 and M2 can be multiplied; (SRF3) rk M1 M2

  • = rk(M1) + rk(M2);

(SRF4) rk M1 M3 M2

  • ≥ rk(M1) + rk(M2) if M1, M2 and

M3 are of appropriate sizes. The space P(R) of Sylvester rank functions on R is a compact convex subset of RMat(R)

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 13

Sylvester rank function on a K-algebra

Let R be a K-algebra. A Sylvester matrix rank function rk on R is a map rk : Mat(R) → R≥0 satisfying the following conditions (SRF1) rk(M) = 0 if M is any zero matrix and rk(1R) = 1; (SRF2) rk(M1M2) ≤ min{rk(M1), rk(M2)} if M1 and M2 can be multiplied; (SRF3) rk M1 M2

  • = rk(M1) + rk(M2);

(SRF4) rk M1 M3 M2

  • ≥ rk(M1) + rk(M2) if M1, M2 and

M3 are of appropriate sizes. The space P(R) of Sylvester rank functions on R is a compact convex subset of RMat(R)

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 14

Sylvester rank function on a K-algebra

Let R be a K-algebra. A Sylvester matrix rank function rk on R is a map rk : Mat(R) → R≥0 satisfying the following conditions (SRF1) rk(M) = 0 if M is any zero matrix and rk(1R) = 1; (SRF2) rk(M1M2) ≤ min{rk(M1), rk(M2)} if M1 and M2 can be multiplied; (SRF3) rk M1 M2

  • = rk(M1) + rk(M2);

(SRF4) rk M1 M3 M2

  • ≥ rk(M1) + rk(M2) if M1, M2 and

M3 are of appropriate sizes. The space P(R) of Sylvester rank functions on R is a compact convex subset of RMat(R)

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 15

The main questions

Main questiones

1 Is there the limit limi→∞ rkG/Gi(A)? 2 If the limit exists, how does it depend on the chain {Gi}? 3 What is the range of possible values for limi→∞ rkG/Gi(A) for

a given group G? Conjectures

1 Yes, the limit exists. 2 It does not depend on the chain {Gi}. 3 Assume that there exists an upper bound for the orders of

finite subgroups of G. Let lcm(G) = lcm{|H| : H is a finite subgroup of G}. Then limi→∞ rkG/Gi(A) ∈

1 lcm(G)Z.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 16

The main questions

Main questiones

1 Is there the limit limi→∞ rkG/Gi(A)? 2 If the limit exists, how does it depend on the chain {Gi}? 3 What is the range of possible values for limi→∞ rkG/Gi(A) for

a given group G? Conjectures

1 Yes, the limit exists. 2 It does not depend on the chain {Gi}. 3 Assume that there exists an upper bound for the orders of

finite subgroups of G. Let lcm(G) = lcm{|H| : H is a finite subgroup of G}. Then limi→∞ rkG/Gi(A) ∈

1 lcm(G)Z.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 17

The main questions

Main questiones

1 Is there the limit limi→∞ rkG/Gi(A)? 2 If the limit exists, how does it depend on the chain {Gi}? 3 What is the range of possible values for limi→∞ rkG/Gi(A) for

a given group G? Conjectures

1 Yes, the limit exists. 2 It does not depend on the chain {Gi}. 3 Assume that there exists an upper bound for the orders of

finite subgroups of G. Let lcm(G) = lcm{|H| : H is a finite subgroup of G}. Then limi→∞ rkG/Gi(A) ∈

1 lcm(G)Z.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 18

The main questions

Main questiones

1 Is there the limit limi→∞ rkG/Gi(A)? 2 If the limit exists, how does it depend on the chain {Gi}? 3 What is the range of possible values for limi→∞ rkG/Gi(A) for

a given group G? Conjectures

1 Yes, the limit exists. 2 It does not depend on the chain {Gi}. 3 Assume that there exists an upper bound for the orders of

finite subgroups of G. Let lcm(G) = lcm{|H| : H is a finite subgroup of G}. Then limi→∞ rkG/Gi(A) ∈

1 lcm(G)Z.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 19

The main questions

Main questiones

1 Is there the limit limi→∞ rkG/Gi(A)? 2 If the limit exists, how does it depend on the chain {Gi}? 3 What is the range of possible values for limi→∞ rkG/Gi(A) for

a given group G? Conjectures

1 Yes, the limit exists. 2 It does not depend on the chain {Gi}. 3 Assume that there exists an upper bound for the orders of

finite subgroups of G. Let lcm(G) = lcm{|H| : H is a finite subgroup of G}. Then limi→∞ rkG/Gi(A) ∈

1 lcm(G)Z.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 20

The main questions

Main questiones

1 Is there the limit limi→∞ rkG/Gi(A)? 2 If the limit exists, how does it depend on the chain {Gi}? 3 What is the range of possible values for limi→∞ rkG/Gi(A) for

a given group G? Conjectures

1 Yes, the limit exists. 2 It does not depend on the chain {Gi}. 3 Assume that there exists an upper bound for the orders of

finite subgroups of G. Let lcm(G) = lcm{|H| : H is a finite subgroup of G}. Then limi→∞ rkG/Gi(A) ∈

1 lcm(G)Z.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 21

The main questions

Main questiones

1 Is there the limit limi→∞ rkG/Gi(A)? 2 If the limit exists, how does it depend on the chain {Gi}? 3 What is the range of possible values for limi→∞ rkG/Gi(A) for

a given group G? Conjectures

1 Yes, the limit exists. 2 It does not depend on the chain {Gi}. 3 Assume that there exists an upper bound for the orders of

finite subgroups of G. Let lcm(G) = lcm{|H| : H is a finite subgroup of G}. Then limi→∞ rkG/Gi(A) ∈

1 lcm(G)Z.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 22

Motivation: Kaplansky’s zero-divisor conjecture

Kaplansky’s zero-divisor conjecture Let G be a torsion-free group. Then the group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Conjectures 1 and 3 predict that if G is torsion-free (lcm(G) = 1), then lim

i→∞ rkG/Gi(A) ∈ Z.

rk = lim

i→∞ rkG/Gi ∈ P(K[G]) is faithful (rk(A) = 0 iff A = 0)

  • P. Cohn: Assume that a K-algebra R has a faithful Sylvester

matrix rank function taking only integer values. Then R can be embedded in a skew field. Thus, Conjectures 1 and 3 imply Kaplansky’s zero-divisor conjecture for K[G]

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 23

Motivation: Kaplansky’s zero-divisor conjecture

Kaplansky’s zero-divisor conjecture Let G be a torsion-free group. Then the group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Conjectures 1 and 3 predict that if G is torsion-free (lcm(G) = 1), then lim

i→∞ rkG/Gi(A) ∈ Z.

rk = lim

i→∞ rkG/Gi ∈ P(K[G]) is faithful (rk(A) = 0 iff A = 0)

  • P. Cohn: Assume that a K-algebra R has a faithful Sylvester

matrix rank function taking only integer values. Then R can be embedded in a skew field. Thus, Conjectures 1 and 3 imply Kaplansky’s zero-divisor conjecture for K[G]

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 24

Motivation: Kaplansky’s zero-divisor conjecture

Kaplansky’s zero-divisor conjecture Let G be a torsion-free group. Then the group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Conjectures 1 and 3 predict that if G is torsion-free (lcm(G) = 1), then lim

i→∞ rkG/Gi(A) ∈ Z.

rk = lim

i→∞ rkG/Gi ∈ P(K[G]) is faithful (rk(A) = 0 iff A = 0)

  • P. Cohn: Assume that a K-algebra R has a faithful Sylvester

matrix rank function taking only integer values. Then R can be embedded in a skew field. Thus, Conjectures 1 and 3 imply Kaplansky’s zero-divisor conjecture for K[G]

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 25

Motivation: Kaplansky’s zero-divisor conjecture

Kaplansky’s zero-divisor conjecture Let G be a torsion-free group. Then the group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Conjectures 1 and 3 predict that if G is torsion-free (lcm(G) = 1), then lim

i→∞ rkG/Gi(A) ∈ Z.

rk = lim

i→∞ rkG/Gi ∈ P(K[G]) is faithful (rk(A) = 0 iff A = 0)

  • P. Cohn: Assume that a K-algebra R has a faithful Sylvester

matrix rank function taking only integer values. Then R can be embedded in a skew field. Thus, Conjectures 1 and 3 imply Kaplansky’s zero-divisor conjecture for K[G]

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 26

Motivation: Kaplansky’s zero-divisor conjecture

Kaplansky’s zero-divisor conjecture Let G be a torsion-free group. Then the group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Conjectures 1 and 3 predict that if G is torsion-free (lcm(G) = 1), then lim

i→∞ rkG/Gi(A) ∈ Z.

rk = lim

i→∞ rkG/Gi ∈ P(K[G]) is faithful (rk(A) = 0 iff A = 0)

  • P. Cohn: Assume that a K-algebra R has a faithful Sylvester

matrix rank function taking only integer values. Then R can be embedded in a skew field. Thus, Conjectures 1 and 3 imply Kaplansky’s zero-divisor conjecture for K[G]

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 27

Motivation: the growth of the Betti numbers in a chain of coverings

Let C be a CW -complex of dimension n and Ci (0 ≤ i ≤ n) its i-dimensional cells. Assume that G acts freely on C conserving the CW -structure and G\C has only a finite number of cells. Goal: we want to analyze lim

i→∞

dimKHp(Gi\C, K) |G : Gi| . We use the cellular chain complex . . . Z[Cp+1]

dp+1

→ Z[Cp]

dp

→ Z[Cp−1] . . . → Z → 0

  • . . .

Z[G]np+1

×Ap+1

→ Z[G]np

×Ap

→ Z[G]np−1 . . . → Z → 0 lim

i→∞

dimKHp(Gi\C, K) |G : Gi| = np − lim

i→∞(rkG/Gi(Ap) + rkG/Gi(Ap+1)).

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 28

Motivation: the growth of the Betti numbers in a chain of coverings

Let C be a CW -complex of dimension n and Ci (0 ≤ i ≤ n) its i-dimensional cells. Assume that G acts freely on C conserving the CW -structure and G\C has only a finite number of cells. Goal: we want to analyze lim

i→∞

dimKHp(Gi\C, K) |G : Gi| . We use the cellular chain complex . . . Z[Cp+1]

dp+1

→ Z[Cp]

dp

→ Z[Cp−1] . . . → Z → 0

  • . . .

Z[G]np+1

×Ap+1

→ Z[G]np

×Ap

→ Z[G]np−1 . . . → Z → 0 lim

i→∞

dimKHp(Gi\C, K) |G : Gi| = np − lim

i→∞(rkG/Gi(Ap) + rkG/Gi(Ap+1)).

Andrei Jaikin-Zapirain L2-Betti numbers

slide-29
SLIDE 29

Motivation: the growth of the Betti numbers in a chain of coverings

Let C be a CW -complex of dimension n and Ci (0 ≤ i ≤ n) its i-dimensional cells. Assume that G acts freely on C conserving the CW -structure and G\C has only a finite number of cells. Goal: we want to analyze lim

i→∞

dimKHp(Gi\C, K) |G : Gi| . We use the cellular chain complex . . . Z[Cp+1]

dp+1

→ Z[Cp]

dp

→ Z[Cp−1] . . . → Z → 0

  • . . .

Z[G]np+1

×Ap+1

→ Z[G]np

×Ap

→ Z[G]np−1 . . . → Z → 0 lim

i→∞

dimKHp(Gi\C, K) |G : Gi| = np − lim

i→∞(rkG/Gi(Ap) + rkG/Gi(Ap+1)).

Andrei Jaikin-Zapirain L2-Betti numbers

slide-30
SLIDE 30

Motivation: the growth of the Betti numbers in a chain of coverings

Let C be a CW -complex of dimension n and Ci (0 ≤ i ≤ n) its i-dimensional cells. Assume that G acts freely on C conserving the CW -structure and G\C has only a finite number of cells. Goal: we want to analyze lim

i→∞

dimKHp(Gi\C, K) |G : Gi| . We use the cellular chain complex . . . Z[Cp+1]

dp+1

→ Z[Cp]

dp

→ Z[Cp−1] . . . → Z → 0

  • . . .

Z[G]np+1

×Ap+1

→ Z[G]np

×Ap

→ Z[G]np−1 . . . → Z → 0 lim

i→∞

dimKHp(Gi\C, K) |G : Gi| = np − lim

i→∞(rkG/Gi(Ap) + rkG/Gi(Ap+1)).

Andrei Jaikin-Zapirain L2-Betti numbers

slide-31
SLIDE 31

Motivation: the growth of the Betti numbers in a chain of coverings

Let C be a CW -complex of dimension n and Ci (0 ≤ i ≤ n) its i-dimensional cells. Assume that G acts freely on C conserving the CW -structure and G\C has only a finite number of cells. Goal: we want to analyze lim

i→∞

dimKHp(Gi\C, K) |G : Gi| . We use the cellular chain complex . . . Z[Cp+1]

dp+1

→ Z[Cp]

dp

→ Z[Cp−1] . . . → Z → 0

  • . . .

Z[G]np+1

×Ap+1

→ Z[G]np

×Ap

→ Z[G]np−1 . . . → Z → 0 lim

i→∞

dimKHp(Gi\C, K) |G : Gi| = np − lim

i→∞(rkG/Gi(Ap) + rkG/Gi(Ap+1)).

Andrei Jaikin-Zapirain L2-Betti numbers

slide-32
SLIDE 32

Motivation: the growth of the Betti numbers in a chain of coverings

Let C be a CW -complex of dimension n and Ci (0 ≤ i ≤ n) its i-dimensional cells. Assume that G acts freely on C conserving the CW -structure and G\C has only a finite number of cells. Goal: we want to analyze lim

i→∞

dimKHp(Gi\C, K) |G : Gi| . We use the cellular chain complex . . . Z[Cp+1]

dp+1

→ Z[Cp]

dp

→ Z[Cp−1] . . . → Z → 0

  • . . .

Z[G]np+1

×Ap+1

→ Z[G]np

×Ap

→ Z[G]np−1 . . . → Z → 0 lim

i→∞

dimKHp(Gi\C, K) |G : Gi| = np − lim

i→∞(rkG/Gi(Ap) + rkG/Gi(Ap+1)).

Andrei Jaikin-Zapirain L2-Betti numbers

slide-33
SLIDE 33

Motivation: the congruence kernel of arithmetic subgroups in SL2(C)

Let Γ be an arithmetic subgroup of SL2(C) (e.g. Γ = SL2(Z[i])). A congruence subgroup of Γ is a subgroup containing Γ(n) = {g ∈ Γ : g ≡ Id (mod n)}. 1 → KΓ → Γ → Γ

congr → 1.

  • A. Lubotzky: Γ does not satisfy weak congruence property
  • F. Grunewald, A. Pinto, A. Jaikin, P. Zalesskii: cd(KΓ) ≤ 2 and

cd(KΓ) = 1 if for every prime p and for every subgroup G ≤f Γ lim

i→∞

dimFp H1(G(pi), Fp) |G : G(pi)| = 0. We know that there exists normal chain G = G1 >f G2 >f . . . lim

i→∞

dimFp H1(Gi, Fp) |G : Gi| = 0.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 34

Motivation: the congruence kernel of arithmetic subgroups in SL2(C)

Let Γ be an arithmetic subgroup of SL2(C) (e.g. Γ = SL2(Z[i])). A congruence subgroup of Γ is a subgroup containing Γ(n) = {g ∈ Γ : g ≡ Id (mod n)}. 1 → KΓ → Γ → Γ

congr → 1.

  • A. Lubotzky: Γ does not satisfy weak congruence property
  • F. Grunewald, A. Pinto, A. Jaikin, P. Zalesskii: cd(KΓ) ≤ 2 and

cd(KΓ) = 1 if for every prime p and for every subgroup G ≤f Γ lim

i→∞

dimFp H1(G(pi), Fp) |G : G(pi)| = 0. We know that there exists normal chain G = G1 >f G2 >f . . . lim

i→∞

dimFp H1(Gi, Fp) |G : Gi| = 0.

Andrei Jaikin-Zapirain L2-Betti numbers

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SLIDE 35

Motivation: the congruence kernel of arithmetic subgroups in SL2(C)

Let Γ be an arithmetic subgroup of SL2(C) (e.g. Γ = SL2(Z[i])). A congruence subgroup of Γ is a subgroup containing Γ(n) = {g ∈ Γ : g ≡ Id (mod n)}. 1 → KΓ → Γ → Γ

congr → 1.

  • A. Lubotzky: Γ does not satisfy weak congruence property
  • F. Grunewald, A. Pinto, A. Jaikin, P. Zalesskii: cd(KΓ) ≤ 2 and

cd(KΓ) = 1 if for every prime p and for every subgroup G ≤f Γ lim

i→∞

dimFp H1(G(pi), Fp) |G : G(pi)| = 0. We know that there exists normal chain G = G1 >f G2 >f . . . lim

i→∞

dimFp H1(Gi, Fp) |G : Gi| = 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-36
SLIDE 36

Motivation: the congruence kernel of arithmetic subgroups in SL2(C)

Let Γ be an arithmetic subgroup of SL2(C) (e.g. Γ = SL2(Z[i])). A congruence subgroup of Γ is a subgroup containing Γ(n) = {g ∈ Γ : g ≡ Id (mod n)}. 1 → KΓ → Γ → Γ

congr → 1.

  • A. Lubotzky: Γ does not satisfy weak congruence property
  • F. Grunewald, A. Pinto, A. Jaikin, P. Zalesskii: cd(KΓ) ≤ 2 and

cd(KΓ) = 1 if for every prime p and for every subgroup G ≤f Γ lim

i→∞

dimFp H1(G(pi), Fp) |G : G(pi)| = 0. We know that there exists normal chain G = G1 >f G2 >f . . . lim

i→∞

dimFp H1(Gi, Fp) |G : Gi| = 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-37
SLIDE 37

Motivation: the congruence kernel of arithmetic subgroups in SL2(C)

Let Γ be an arithmetic subgroup of SL2(C) (e.g. Γ = SL2(Z[i])). A congruence subgroup of Γ is a subgroup containing Γ(n) = {g ∈ Γ : g ≡ Id (mod n)}. 1 → KΓ → Γ → Γ

congr → 1.

  • A. Lubotzky: Γ does not satisfy weak congruence property
  • F. Grunewald, A. Pinto, A. Jaikin, P. Zalesskii: cd(KΓ) ≤ 2 and

cd(KΓ) = 1 if for every prime p and for every subgroup G ≤f Γ lim

i→∞

dimFp H1(G(pi), Fp) |G : G(pi)| = 0. We know that there exists normal chain G = G1 >f G2 >f . . . lim

i→∞

dimFp H1(Gi, Fp) |G : Gi| = 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-38
SLIDE 38

Motivation: the congruence kernel of arithmetic subgroups in SL2(C)

Let Γ be an arithmetic subgroup of SL2(C) (e.g. Γ = SL2(Z[i])). A congruence subgroup of Γ is a subgroup containing Γ(n) = {g ∈ Γ : g ≡ Id (mod n)}. 1 → KΓ → Γ → Γ

congr → 1.

  • A. Lubotzky: Γ does not satisfy weak congruence property
  • F. Grunewald, A. Pinto, A. Jaikin, P. Zalesskii: cd(KΓ) ≤ 2 and

cd(KΓ) = 1 if for every prime p and for every subgroup G ≤f Γ lim

i→∞

dimFp H1(G(pi), Fp) |G : G(pi)| = 0. We know that there exists normal chain G = G1 >f G2 >f . . . lim

i→∞

dimFp H1(Gi, Fp) |G : Gi| = 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-39
SLIDE 39

Motivation: the congruence kernel of arithmetic subgroups in SL2(C)

Let Γ be an arithmetic subgroup of SL2(C) (e.g. Γ = SL2(Z[i])). A congruence subgroup of Γ is a subgroup containing Γ(n) = {g ∈ Γ : g ≡ Id (mod n)}. 1 → KΓ → Γ → Γ

congr → 1.

  • A. Lubotzky: Γ does not satisfy weak congruence property
  • F. Grunewald, A. Pinto, A. Jaikin, P. Zalesskii: cd(KΓ) ≤ 2 and

cd(KΓ) = 1 if for every prime p and for every subgroup G ≤f Γ lim

i→∞

dimFp H1(G(pi), Fp) |G : G(pi)| = 0. We know that there exists normal chain G = G1 >f G2 >f . . . lim

i→∞

dimFp H1(Gi, Fp) |G : Gi| = 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-40
SLIDE 40

Motivation: the Hanna Neumann conjecture

F is a free group; U and W are f.g. subgroups of F; rk(U) = max{rk(U) − 1, 0}. 1957: H. Neumann: rk(U ∩ W ) ≤ 2rk(U)rk(W ) The Hanna Neumann conjecture: rk(U ∩ W ) ≤ rk(U)rk(W ) 1990: W. Neumann:

x∈U\F/W rk(U ∩ xWx−1) ≤ 2rk(U)rk(W ).

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

Solved independently by J. Friedman (2011) and I. Mineyev (2011)

Andrei Jaikin-Zapirain L2-Betti numbers

slide-41
SLIDE 41

Motivation: the Hanna Neumann conjecture

F is a free group; U and W are f.g. subgroups of F; rk(U) = max{rk(U) − 1, 0}. 1957: H. Neumann: rk(U ∩ W ) ≤ 2rk(U)rk(W ) The Hanna Neumann conjecture: rk(U ∩ W ) ≤ rk(U)rk(W ) 1990: W. Neumann:

x∈U\F/W rk(U ∩ xWx−1) ≤ 2rk(U)rk(W ).

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

Solved independently by J. Friedman (2011) and I. Mineyev (2011)

Andrei Jaikin-Zapirain L2-Betti numbers

slide-42
SLIDE 42

Motivation: the Hanna Neumann conjecture

F is a free group; U and W are f.g. subgroups of F; rk(U) = max{rk(U) − 1, 0}. 1957: H. Neumann: rk(U ∩ W ) ≤ 2rk(U)rk(W ) The Hanna Neumann conjecture: rk(U ∩ W ) ≤ rk(U)rk(W ) 1990: W. Neumann:

x∈U\F/W rk(U ∩ xWx−1) ≤ 2rk(U)rk(W ).

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

Solved independently by J. Friedman (2011) and I. Mineyev (2011)

Andrei Jaikin-Zapirain L2-Betti numbers

slide-43
SLIDE 43

Motivation: the Hanna Neumann conjecture

F is a free group; U and W are f.g. subgroups of F; rk(U) = max{rk(U) − 1, 0}. 1957: H. Neumann: rk(U ∩ W ) ≤ 2rk(U)rk(W ) The Hanna Neumann conjecture: rk(U ∩ W ) ≤ rk(U)rk(W ) 1990: W. Neumann:

x∈U\F/W rk(U ∩ xWx−1) ≤ 2rk(U)rk(W ).

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

Solved independently by J. Friedman (2011) and I. Mineyev (2011)

Andrei Jaikin-Zapirain L2-Betti numbers

slide-44
SLIDE 44

Motivation: the Hanna Neumann conjecture

F is a free group; U and W are f.g. subgroups of F; rk(U) = max{rk(U) − 1, 0}. 1957: H. Neumann: rk(U ∩ W ) ≤ 2rk(U)rk(W ) The Hanna Neumann conjecture: rk(U ∩ W ) ≤ rk(U)rk(W ) 1990: W. Neumann:

x∈U\F/W rk(U ∩ xWx−1) ≤ 2rk(U)rk(W ).

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

Solved independently by J. Friedman (2011) and I. Mineyev (2011)

Andrei Jaikin-Zapirain L2-Betti numbers

slide-45
SLIDE 45

Motivation: the Hanna Neumann conjecture

F is a free group; U and W are f.g. subgroups of F; rk(U) = max{rk(U) − 1, 0}. 1957: H. Neumann: rk(U ∩ W ) ≤ 2rk(U)rk(W ) The Hanna Neumann conjecture: rk(U ∩ W ) ≤ rk(U)rk(W ) 1990: W. Neumann:

x∈U\F/W rk(U ∩ xWx−1) ≤ 2rk(U)rk(W ).

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

Solved independently by J. Friedman (2011) and I. Mineyev (2011)

Andrei Jaikin-Zapirain L2-Betti numbers

slide-46
SLIDE 46

Motivation: the Hanna Neumann conjecture

F is a free group; U and W are f.g. subgroups of F; rk(U) = max{rk(U) − 1, 0}. 1957: H. Neumann: rk(U ∩ W ) ≤ 2rk(U)rk(W ) The Hanna Neumann conjecture: rk(U ∩ W ) ≤ rk(U)rk(W ) 1990: W. Neumann:

x∈U\F/W rk(U ∩ xWx−1) ≤ 2rk(U)rk(W ).

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

Solved independently by J. Friedman (2011) and I. Mineyev (2011)

Andrei Jaikin-Zapirain L2-Betti numbers

slide-47
SLIDE 47

Motivation: the Hanna Neumann conjecture

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

2015: A. Jaikin-Zapirain: The strengthened Hanna Neumann conjecture for free pro-p groups and a new proof for free groups This new proof uses the strong Atiyah and the L¨ uck approximation conjectures over C. 2017: Y. Antolin, A. Jaikin-Zapirain: The strengthened Hanna Neumann conjecture for non-abelian surface groups.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-48
SLIDE 48

Motivation: the Hanna Neumann conjecture

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

2015: A. Jaikin-Zapirain: The strengthened Hanna Neumann conjecture for free pro-p groups and a new proof for free groups This new proof uses the strong Atiyah and the L¨ uck approximation conjectures over C. 2017: Y. Antolin, A. Jaikin-Zapirain: The strengthened Hanna Neumann conjecture for non-abelian surface groups.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-49
SLIDE 49

Motivation: the Hanna Neumann conjecture

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

2015: A. Jaikin-Zapirain: The strengthened Hanna Neumann conjecture for free pro-p groups and a new proof for free groups This new proof uses the strong Atiyah and the L¨ uck approximation conjectures over C. 2017: Y. Antolin, A. Jaikin-Zapirain: The strengthened Hanna Neumann conjecture for non-abelian surface groups.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-50
SLIDE 50

Motivation: the Hanna Neumann conjecture

The strengthened Hanna Neumann conjecture:

  • x∈U\F/W rk(U ∩ xWx−1) ≤ rk(U)rk(W ).

2015: A. Jaikin-Zapirain: The strengthened Hanna Neumann conjecture for free pro-p groups and a new proof for free groups This new proof uses the strong Atiyah and the L¨ uck approximation conjectures over C. 2017: Y. Antolin, A. Jaikin-Zapirain: The strengthened Hanna Neumann conjecture for non-abelian surface groups.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-51
SLIDE 51

The dimension of Hilbert G-modules

A Hilbert G-module V is a closed (left G)-invariant subspace of the Hilbert space l2(G)n: dimG V = n

k=1projV (ek), ek

G is finite: l2(G) = C[G] and dimG V = dimC V

|G| .

If A ∈ Matn×m(C[G]) and N G, we put φA

G/N : l2(G/N)n → l2(G/N)m ; φA G/N(v1, . . . , vn) = (v1, . . . , vn)A

rkG/N(A) := dimG/N(Im φA

G/N) = n − dimG/N ker φA G/N.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-52
SLIDE 52

The dimension of Hilbert G-modules

A Hilbert G-module V is a closed (left G)-invariant subspace of the Hilbert space l2(G)n: dimG V = n

k=1projV (ek), ek

G is finite: l2(G) = C[G] and dimG V = dimC V

|G| .

If A ∈ Matn×m(C[G]) and N G, we put φA

G/N : l2(G/N)n → l2(G/N)m ; φA G/N(v1, . . . , vn) = (v1, . . . , vn)A

rkG/N(A) := dimG/N(Im φA

G/N) = n − dimG/N ker φA G/N.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-53
SLIDE 53

The dimension of Hilbert G-modules

A Hilbert G-module V is a closed (left G)-invariant subspace of the Hilbert space l2(G)n: dimG V = n

k=1projV (ek), ek

G is finite: l2(G) = C[G] and dimG V = dimC V

|G| .

If A ∈ Matn×m(C[G]) and N G, we put φA

G/N : l2(G/N)n → l2(G/N)m ; φA G/N(v1, . . . , vn) = (v1, . . . , vn)A

rkG/N(A) := dimG/N(Im φA

G/N) = n − dimG/N ker φA G/N.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-54
SLIDE 54

The dimension of Hilbert G-modules

A Hilbert G-module V is a closed (left G)-invariant subspace of the Hilbert space l2(G)n: dimG V = n

k=1projV (ek), ek

G is finite: l2(G) = C[G] and dimG V = dimC V

|G| .

If A ∈ Matn×m(C[G]) and N G, we put φA

G/N : l2(G/N)n → l2(G/N)m ; φA G/N(v1, . . . , vn) = (v1, . . . , vn)A

rkG/N(A) := dimG/N(Im φA

G/N) = n − dimG/N ker φA G/N.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-55
SLIDE 55

The L¨ uck approximation and the strong Atiyah conjectures

Let K be a subfield of C Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Conjectures (with coefficients in K) L (the L¨ uck approximation conjecture over K) For every matrix A over K[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A (the strong Atiyah conjecture over K) Assume that there exists an upper bound for the orders of finite subgroups of G. For every matrix A over K[G], rkG(A) ∈

1 lcm(G)Z.

K ≤ C: Conjecture L ⇒ Conjectures 1 and 2 Conjectures L and A ⇒ Conjecture 3

Andrei Jaikin-Zapirain L2-Betti numbers

slide-56
SLIDE 56

The L¨ uck approximation and the strong Atiyah conjectures

Let K be a subfield of C Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Conjectures (with coefficients in K) L (the L¨ uck approximation conjecture over K) For every matrix A over K[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A (the strong Atiyah conjecture over K) Assume that there exists an upper bound for the orders of finite subgroups of G. For every matrix A over K[G], rkG(A) ∈

1 lcm(G)Z.

K ≤ C: Conjecture L ⇒ Conjectures 1 and 2 Conjectures L and A ⇒ Conjecture 3

Andrei Jaikin-Zapirain L2-Betti numbers

slide-57
SLIDE 57

The L¨ uck approximation and the strong Atiyah conjectures

Let K be a subfield of C Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Conjectures (with coefficients in K) L (the L¨ uck approximation conjecture over K) For every matrix A over K[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A (the strong Atiyah conjecture over K) Assume that there exists an upper bound for the orders of finite subgroups of G. For every matrix A over K[G], rkG(A) ∈

1 lcm(G)Z.

K ≤ C: Conjecture L ⇒ Conjectures 1 and 2 Conjectures L and A ⇒ Conjecture 3

Andrei Jaikin-Zapirain L2-Betti numbers

slide-58
SLIDE 58

The L¨ uck approximation and the strong Atiyah conjectures

Let K be a subfield of C Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Conjectures (with coefficients in K) L (the L¨ uck approximation conjecture over K) For every matrix A over K[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A (the strong Atiyah conjecture over K) Assume that there exists an upper bound for the orders of finite subgroups of G. For every matrix A over K[G], rkG(A) ∈

1 lcm(G)Z.

K ≤ C: Conjecture L ⇒ Conjectures 1 and 2 Conjectures L and A ⇒ Conjecture 3

Andrei Jaikin-Zapirain L2-Betti numbers

slide-59
SLIDE 59

The L¨ uck approximation and the strong Atiyah conjectures

Let K be a subfield of C Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Conjectures (with coefficients in K) L (the L¨ uck approximation conjecture over K) For every matrix A over K[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A (the strong Atiyah conjecture over K) Assume that there exists an upper bound for the orders of finite subgroups of G. For every matrix A over K[G], rkG(A) ∈

1 lcm(G)Z.

K ≤ C: Conjecture L ⇒ Conjectures 1 and 2 Conjectures L and A ⇒ Conjecture 3

Andrei Jaikin-Zapirain L2-Betti numbers

slide-60
SLIDE 60

The L¨ uck approximation and the strong Atiyah conjectures

Let K be a subfield of C Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Conjectures (with coefficients in K) L (the L¨ uck approximation conjecture over K) For every matrix A over K[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A (the strong Atiyah conjecture over K) Assume that there exists an upper bound for the orders of finite subgroups of G. For every matrix A over K[G], rkG(A) ∈

1 lcm(G)Z.

K ≤ C: Conjecture L ⇒ Conjectures 1 and 2 Conjectures L and A ⇒ Conjecture 3

Andrei Jaikin-Zapirain L2-Betti numbers

slide-61
SLIDE 61

The state of the conjectures

The class of elementary amenable groups is the smallest class of groups containing finite groups, abelian groups and closed under subgroups, extensions and direct unions. K ≤ C charK > 0

  • Conj. 1

Yes Yes

  • Conj. 2

Yes Yes

  • Conj. 3

Yes Yes

  • Conj. L

Yes X

  • Conj. A

Yes X

Andrei Jaikin-Zapirain L2-Betti numbers

slide-62
SLIDE 62

The state of the conjectures

The class of elementary amenable groups is the smallest class of groups containing finite groups, abelian groups and closed under subgroups, extensions and direct unions. K ≤ C charK > 0

  • Conj. 1

Yes Yes

  • Conj. 2

Yes Yes

  • Conj. 3

Yes Yes

  • Conj. L

Yes X

  • Conj. A

Yes X

Andrei Jaikin-Zapirain L2-Betti numbers

slide-63
SLIDE 63

The state of the conjectures

A finitely generated group G is amenable if there exists a family {Fi} of finite subsets of G such that for any g ∈ G lim

i→∞

|gFi ∩ Fi| |Fi| = 1. K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes

  • Conj. 2

Yes Yes Yes Yes

  • Conj. 3

Yes ? Yes ?

  • Conj. L

Yes Yes X

  • Conj. A

Yes ? X elemantary amenable amenable

Andrei Jaikin-Zapirain L2-Betti numbers

slide-64
SLIDE 64

The state of the conjectures

A finitely generated group G is amenable if there exists a family {Fi} of finite subsets of G such that for any g ∈ G lim

i→∞

|gFi ∩ Fi| |Fi| = 1. K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes

  • Conj. 2

Yes Yes Yes Yes

  • Conj. 3

Yes ? Yes ?

  • Conj. L

Yes Yes X

  • Conj. A

Yes ? X elemantary amenable amenable

Andrei Jaikin-Zapirain L2-Betti numbers

slide-65
SLIDE 65

The state of the conjectures

residually torsion-free soluble groups; hyperbolic 3-orbifold groups; virtually special groups K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes Yes ?

  • Conj. 2

Yes Yes Yes Yes Yes ?

  • Conj. 3

Yes ? Yes Yes ? ?

  • Conj. L

Yes Yes Yes X

  • Conj. A

Yes ? Yes X elemantary amenable; amenable

Andrei Jaikin-Zapirain L2-Betti numbers

slide-66
SLIDE 66

The state of the conjectures

residually torsion-free soluble groups; hyperbolic 3-orbifold groups; virtually special groups K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes Yes ?

  • Conj. 2

Yes Yes Yes Yes Yes ?

  • Conj. 3

Yes ? Yes Yes ? ?

  • Conj. L

Yes Yes Yes X

  • Conj. A

Yes ? Yes X elemantary amenable; amenable

Andrei Jaikin-Zapirain L2-Betti numbers

slide-67
SLIDE 67

The state of the conjectures

residually torsion-free soluble groups; hyperbolic 3-orbifold groups; virtually special groups K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes Yes ?

  • Conj. 2

Yes Yes Yes Yes Yes ?

  • Conj. 3

Yes ? Yes Yes ? ?

  • Conj. L

Yes Yes Yes X

  • Conj. A

Yes ? Yes X elemantary amenable; amenable

Andrei Jaikin-Zapirain L2-Betti numbers

slide-68
SLIDE 68

The state of the conjectures

residually torsion-free soluble groups; hyperbolic 3-orbifold groups; virtually special groups K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes Yes ?

  • Conj. 2

Yes Yes Yes Yes Yes ?

  • Conj. 3

Yes ? Yes Yes ? ?

  • Conj. L

Yes Yes Yes X

  • Conj. A

Yes ? Yes X elemantary amenable; amenable

Andrei Jaikin-Zapirain L2-Betti numbers

slide-69
SLIDE 69

The state of the conjectures

A finitely generated group G = S is sofic if for any ǫ > 0 and for any k there exists a finite S-labeled graph X = (V , E) such that for at least (1 − ǫ)|V | vertices v ∈ V of X, Bk(v) is isomorphic (as a S-labeled graph) to Bk(1G) (a ball in the Cayley graph Cay(G, S)). amenable and residually finite groups are sofic K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes Yes Yes ? ?

  • Conj. 2

Yes Yes Yes Yes Yes Yes ? ?

  • Conj. 3

Yes ? Yes ? Yes ? ? ?

  • Conj. L

Yes Yes Yes Yes X

  • Conj. A

Yes ? Yes ? X elemantary amenable; amenable; residually torsion-free soluble; hyperbolic 3-orbifold; virtually special

Andrei Jaikin-Zapirain L2-Betti numbers

slide-70
SLIDE 70

The state of the conjectures

A finitely generated group G = S is sofic if for any ǫ > 0 and for any k there exists a finite S-labeled graph X = (V , E) such that for at least (1 − ǫ)|V | vertices v ∈ V of X, Bk(v) is isomorphic (as a S-labeled graph) to Bk(1G) (a ball in the Cayley graph Cay(G, S)). amenable and residually finite groups are sofic K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes Yes Yes ? ?

  • Conj. 2

Yes Yes Yes Yes Yes Yes ? ?

  • Conj. 3

Yes ? Yes ? Yes ? ? ?

  • Conj. L

Yes Yes Yes Yes X

  • Conj. A

Yes ? Yes ? X elemantary amenable; amenable; residually torsion-free soluble; hyperbolic 3-orbifold; virtually special

Andrei Jaikin-Zapirain L2-Betti numbers

slide-71
SLIDE 71

The state of the conjectures

A finitely generated group G = S is sofic if for any ǫ > 0 and for any k there exists a finite S-labeled graph X = (V , E) such that for at least (1 − ǫ)|V | vertices v ∈ V of X, Bk(v) is isomorphic (as a S-labeled graph) to Bk(1G) (a ball in the Cayley graph Cay(G, S)). amenable and residually finite groups are sofic K ≤ C charK > 0

  • Conj. 1

Yes Yes Yes Yes Yes Yes ? ?

  • Conj. 2

Yes Yes Yes Yes Yes Yes ? ?

  • Conj. 3

Yes ? Yes ? Yes ? ? ?

  • Conj. L

Yes Yes Yes Yes X

  • Conj. A

Yes ? Yes ? X elemantary amenable; amenable; residually torsion-free soluble; hyperbolic 3-orbifold; virtually special

Andrei Jaikin-Zapirain L2-Betti numbers

slide-72
SLIDE 72

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-73
SLIDE 73

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-74
SLIDE 74

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-75
SLIDE 75

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-76
SLIDE 76

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-77
SLIDE 77

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-78
SLIDE 78

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-79
SLIDE 79

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-80
SLIDE 80

The state of the conjectures

K ≤ ¯ Q K ≤ C charK > 0

  • Conj. 1

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 2

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 Yes2 Yes2 ? ?

  • Conj. 3

Yes1 ? Yes4 ? Yes1 ? Yes5 ? Yes1 ? ? ?

  • Conj. L

Yes2 Yes2 Yes3 Yes3 Yes2 Yes2 Yes6 Yes6 X

  • Conj. A

Yes1 ? Yes4 ? Yes1 ? Yes5 ? X elemantary amenable; amenable; virtually special; sofic

  • M. Atiyah (1974), J. Dodziuk (1977)

1 J. Moody (1987), P. Linnell (1993), G. Elek (2006) 2 G. Elek (2006) 3 W. L¨

uck (1993) ({G/Gi} finite and K = Q),

  • J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates (2003),
  • G. Elek, E. Szabo (2005)

4 [DLMSY], P. Linnell, T. Schick (2007), K. Schreve (2014) 5 P. Linnell (1993) (free, surface groups), A. Jaikin-Zapirain

(2016)

6 A. Jaikin-Zapirain (2017) Andrei Jaikin-Zapirain L2-Betti numbers

slide-81
SLIDE 81

The ideas of the proofs: an analytic approach.

Theorem (L¨ uck (1993), [DLMSY, 2003], G. Elek, E. Szabo (2005)) Let G be a group and let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over ¯ Q[G], limi→∞ rkG/Gi(A) = rkG(A).

1 We may assume that A = BB∗, whence φA

G/Gi and φA G are

selfadjoint positive operators

2 We associate measures µA

G/Gi with φA G/Gi on an interval [0, a].

The theorem is equivalent to show that lim

i→∞ µA G/Gi(0) = µA G(0)

3 µA

G/Gi weakly converges to µA G, whence,

lim sup

i→∞

µA

G/Gi(0) ≤ µA G(0).

4 We use the conditions G/Gi are sofic and K ≤ ¯

Q to show that µA

G/Gi(0, ǫ) tends uniformly in i to 0 when ǫ tends to 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-82
SLIDE 82

The ideas of the proofs: an analytic approach.

Theorem (L¨ uck (1993), [DLMSY, 2003], G. Elek, E. Szabo (2005)) Let G be a group and let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over ¯ Q[G], limi→∞ rkG/Gi(A) = rkG(A).

1 We may assume that A = BB∗, whence φA

G/Gi and φA G are

selfadjoint positive operators

2 We associate measures µA

G/Gi with φA G/Gi on an interval [0, a].

The theorem is equivalent to show that lim

i→∞ µA G/Gi(0) = µA G(0)

3 µA

G/Gi weakly converges to µA G, whence,

lim sup

i→∞

µA

G/Gi(0) ≤ µA G(0).

4 We use the conditions G/Gi are sofic and K ≤ ¯

Q to show that µA

G/Gi(0, ǫ) tends uniformly in i to 0 when ǫ tends to 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-83
SLIDE 83

The ideas of the proofs: an analytic approach.

Theorem (L¨ uck (1993), [DLMSY, 2003], G. Elek, E. Szabo (2005)) Let G be a group and let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over ¯ Q[G], limi→∞ rkG/Gi(A) = rkG(A).

1 We may assume that A = BB∗, whence φA

G/Gi and φA G are

selfadjoint positive operators

2 We associate measures µA

G/Gi with φA G/Gi on an interval [0, a].

The theorem is equivalent to show that lim

i→∞ µA G/Gi(0) = µA G(0)

3 µA

G/Gi weakly converges to µA G, whence,

lim sup

i→∞

µA

G/Gi(0) ≤ µA G(0).

4 We use the conditions G/Gi are sofic and K ≤ ¯

Q to show that µA

G/Gi(0, ǫ) tends uniformly in i to 0 when ǫ tends to 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-84
SLIDE 84

The ideas of the proofs: an analytic approach.

Theorem (L¨ uck (1993), [DLMSY, 2003], G. Elek, E. Szabo (2005)) Let G be a group and let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over ¯ Q[G], limi→∞ rkG/Gi(A) = rkG(A).

1 We may assume that A = BB∗, whence φA

G/Gi and φA G are

selfadjoint positive operators

2 We associate measures µA

G/Gi with φA G/Gi on an interval [0, a].

The theorem is equivalent to show that lim

i→∞ µA G/Gi(0) = µA G(0)

3 µA

G/Gi weakly converges to µA G, whence,

lim sup

i→∞

µA

G/Gi(0) ≤ µA G(0).

4 We use the conditions G/Gi are sofic and K ≤ ¯

Q to show that µA

G/Gi(0, ǫ) tends uniformly in i to 0 when ǫ tends to 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-85
SLIDE 85

The ideas of the proofs: an analytic approach.

Theorem (L¨ uck (1993), [DLMSY, 2003], G. Elek, E. Szabo (2005)) Let G be a group and let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over ¯ Q[G], limi→∞ rkG/Gi(A) = rkG(A).

1 We may assume that A = BB∗, whence φA

G/Gi and φA G are

selfadjoint positive operators

2 We associate measures µA

G/Gi with φA G/Gi on an interval [0, a].

The theorem is equivalent to show that lim

i→∞ µA G/Gi(0) = µA G(0)

3 µA

G/Gi weakly converges to µA G, whence,

lim sup

i→∞

µA

G/Gi(0) ≤ µA G(0).

4 We use the conditions G/Gi are sofic and K ≤ ¯

Q to show that µA

G/Gi(0, ǫ) tends uniformly in i to 0 when ǫ tends to 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-86
SLIDE 86

The ideas of the proofs: an analytic approach.

Theorem (L¨ uck (1993), [DLMSY, 2003], G. Elek, E. Szabo (2005)) Let G be a group and let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over ¯ Q[G], limi→∞ rkG/Gi(A) = rkG(A).

1 We may assume that A = BB∗, whence φA

G/Gi and φA G are

selfadjoint positive operators

2 We associate measures µA

G/Gi with φA G/Gi on an interval [0, a].

The theorem is equivalent to show that lim

i→∞ µA G/Gi(0) = µA G(0)

3 µA

G/Gi weakly converges to µA G, whence,

lim sup

i→∞

µA

G/Gi(0) ≤ µA G(0).

4 We use the conditions G/Gi are sofic and K ≤ ¯

Q to show that µA

G/Gi(0, ǫ) tends uniformly in i to 0 when ǫ tends to 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-87
SLIDE 87

The ideas of the proofs: an analytic approach.

Theorem (L¨ uck (1993), [DLMSY, 2003], G. Elek, E. Szabo (2005)) Let G be a group and let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over ¯ Q[G], limi→∞ rkG/Gi(A) = rkG(A).

1 We may assume that A = BB∗, whence φA

G/Gi and φA G are

selfadjoint positive operators

2 We associate measures µA

G/Gi with φA G/Gi on an interval [0, a].

The theorem is equivalent to show that lim

i→∞ µA G/Gi(0) = µA G(0)

3 µA

G/Gi weakly converges to µA G, whence,

lim sup

i→∞

µA

G/Gi(0) ≤ µA G(0).

4 We use the conditions G/Gi are sofic and K ≤ ¯

Q to show that µA

G/Gi(0, ǫ) tends uniformly in i to 0 when ǫ tends to 0.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-88
SLIDE 88

The ideas of the proofs: an algebraic approach.

Theorem (Jaikin (2017)) Let G be a sofic group. L Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over C[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A If G satisfies the strong Atiyah conjecture over ¯ Q, then G satisfies the strong Atiyah conjecture over C.

1 We show that Conjecture L over K is equivalent to the

existence of isomorphism between K[G]-rings RK[G] and RK[G],{G/Gi}.

2 Using that there exists an isomorphism in the case K = Q, we

construct an isomorphism in the case K = C.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-89
SLIDE 89

The ideas of the proofs: an algebraic approach.

Theorem (Jaikin (2017)) Let G be a sofic group. L Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over C[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A If G satisfies the strong Atiyah conjecture over ¯ Q, then G satisfies the strong Atiyah conjecture over C.

1 We show that Conjecture L over K is equivalent to the

existence of isomorphism between K[G]-rings RK[G] and RK[G],{G/Gi}.

2 Using that there exists an isomorphism in the case K = Q, we

construct an isomorphism in the case K = C.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-90
SLIDE 90

The ideas of the proofs: an algebraic approach.

Theorem (Jaikin (2017)) Let G be a sofic group. L Let G = G1 > G2 > . . . be a chain of normal subgroups with trivial intersection. Assume G/Gi are sofic. Then for every matrix A over C[G], lim

i→∞ rkG/Gi(A) = rkG(A).

A If G satisfies the strong Atiyah conjecture over ¯ Q, then G satisfies the strong Atiyah conjecture over C.

1 We show that Conjecture L over K is equivalent to the

existence of isomorphism between K[G]-rings RK[G] and RK[G],{G/Gi}.

2 Using that there exists an isomorphism in the case K = Q, we

construct an isomorphism in the case K = C.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-91
SLIDE 91

Main open problems

Problem 1 Extend the results from the characteristic 0 case to the characteristic p > 0 case. Problem 2 Show that the strong Atiyah conjecture holds for one-relator groups.

  • ne-relator groups with torsion are virtually special

Problem 3 Show that the strong Atiyah conjecture holds for subgroups of GLn(C). If G is a f.g. subgroup of GLn(C) then it is known that there exists H <f G such that H satisfies the strong Atiyah conjecture.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-92
SLIDE 92

Main open problems

Problem 1 Extend the results from the characteristic 0 case to the characteristic p > 0 case. Problem 2 Show that the strong Atiyah conjecture holds for one-relator groups.

  • ne-relator groups with torsion are virtually special

Problem 3 Show that the strong Atiyah conjecture holds for subgroups of GLn(C). If G is a f.g. subgroup of GLn(C) then it is known that there exists H <f G such that H satisfies the strong Atiyah conjecture.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-93
SLIDE 93

Main open problems

Problem 1 Extend the results from the characteristic 0 case to the characteristic p > 0 case. Problem 2 Show that the strong Atiyah conjecture holds for one-relator groups.

  • ne-relator groups with torsion are virtually special

Problem 3 Show that the strong Atiyah conjecture holds for subgroups of GLn(C). If G is a f.g. subgroup of GLn(C) then it is known that there exists H <f G such that H satisfies the strong Atiyah conjecture.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-94
SLIDE 94

Main open problems

Problem 1 Extend the results from the characteristic 0 case to the characteristic p > 0 case. Problem 2 Show that the strong Atiyah conjecture holds for one-relator groups.

  • ne-relator groups with torsion are virtually special

Problem 3 Show that the strong Atiyah conjecture holds for subgroups of GLn(C). If G is a f.g. subgroup of GLn(C) then it is known that there exists H <f G such that H satisfies the strong Atiyah conjecture.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-95
SLIDE 95

Main open problems

Problem 1 Extend the results from the characteristic 0 case to the characteristic p > 0 case. Problem 2 Show that the strong Atiyah conjecture holds for one-relator groups.

  • ne-relator groups with torsion are virtually special

Problem 3 Show that the strong Atiyah conjecture holds for subgroups of GLn(C). If G is a f.g. subgroup of GLn(C) then it is known that there exists H <f G such that H satisfies the strong Atiyah conjecture.

Andrei Jaikin-Zapirain L2-Betti numbers

slide-96
SLIDE 96

Thanks

THANK YOU FOR YOUR ATTENTION

Andrei Jaikin-Zapirain L2-Betti numbers