Kazhdan's Property

Kazhdan’s Property (T) Talk 1: Deﬁnitions, examples and non-examples.

David Hume, University of Bristol [email protected]

October 5, 2020

David Hume Kazhdan’s Property (T) My goal is to demystify some of the deﬁnitions, explain strategies to prove that a group does or doesn’t have property (T), and survey some of the many applications of property (T).

Disclaimer

The book “Kazhdan’s Property (T)” by Bekka, de la Harpe and Valette is an excellent (and freely available) resource.

David Hume Kazhdan’s Property (T) Disclaimer

The book “Kazhdan’s Property (T)” by Bekka, de la Harpe and Valette is an excellent (and freely available) resource. My goal is to demystify some of the deﬁnitions, explain strategies to prove that a group does or doesn’t have property (T), and survey some of the many applications of property (T).

David Hume Kazhdan’s Property (T) Strategy: find a property (T) which satisfies the following three conditions. 1) Simple Lie groups of rank ≥ 2 have property (T). 2) If a locally compact group G has property (T), then all lattices in G have property (T). 3) A discrete countable group which has property (T) admits a finite generating set and has finite abelianisation. To find such a property, Kazhdan considered unitary representations of groups. Why?

Kazhdan’s motivation

Theorem Lattices in simple Lie groups of rank ≥ 2 are ﬁnitely generated and have ﬁnite abelianisation.

David Hume Kazhdan’s Property (T) To ﬁnd such a property, Kazhdan considered unitary representations of groups. Why?

Kazhdan’s motivation

Theorem Lattices in simple Lie groups of rank ≥ 2 are finitely generated and have finite abelianisation. Strategy: find a property (T) which satisfies the following three conditions. 1) Simple Lie groups of rank ≥ 2 have property (T). 2) If a locally compact group G has property (T), then all lattices in G have property (T). 3) A discrete countable group which has property (T) admits a finite generating set and has finite abelianisation.

David Hume Kazhdan’s Property (T) Kazhdan’s motivation

David Hume Kazhdan’s Property (T) The unitary group U(H) of H is the group of all invertible bounded linear operators U ∶ H → H such that

⟨Ux, Uy⟩ = ⟨x, y⟩ for all x, y ∈ H.

A unitary representation of G in H is a group homomorphism π ∶ G → U(H) which is strongly continuous, meaning that for each x ∈ H, the map

G → H given by g ↦ π(g)x is continuous.

Representations

Let G be a locally compact topological group and let H be a complex Hilbert space.

David Hume Kazhdan’s Property (T) A unitary representation of G in H is a group homomorphism π ∶ G → U(H) which is strongly continuous, meaning that for each x ∈ H, the map

G → H given by g ↦ π(g)x is continuous.

Representations

Let G be a locally compact topological group and let H be a complex Hilbert space. The unitary group U(H) of H is the group of all invertible bounded linear operators U ∶ H → H such that

⟨Ux, Uy⟩ = ⟨x, y⟩ for all x, y ∈ H.

David Hume Kazhdan’s Property (T) Representations

Let G be a locally compact topological group and let H be a complex Hilbert space. The unitary group U(H) of H is the group of all invertible bounded linear operators U ∶ H → H such that

⟨Ux, Uy⟩ = ⟨x, y⟩ for all x, y ∈ H.

A unitary representation of G in H is a group homomorphism π ∶ G → U(H) which is strongly continuous, meaning that for each x ∈ H, the map

G → H given by g ↦ π(g)x is continuous.

David Hume Kazhdan’s Property (T) A representation π ∶ G → U(H) is said to contain the trivial representation if it has a non-zero invariant vector. This is denoted 1 ⊂ π. Given a locally compact group G with left invariant Haar measure 2 dg, the left regular representation λG ∶ G → U(L (G, dg)) given by 1 2 λG (g)F (x) = F (g x) for all F ∈ L (G, dg) and x ∈ G is a unitary representation. −

We have 1 ⊂ λG if and only if G is compact.

Examples of Unitary Representations

The trivial representation is deﬁned as 1 ∶ G → C given by 1(g)x = x for all g ∈ G and x ∈ C.

David Hume Kazhdan’s Property (T) Given a locally compact group G with left invariant Haar measure 2 dg, the left regular representation λG ∶ G → U(L (G, dg)) given by 1 2 λG (g)F (x) = F (g x) for all F ∈ L (G, dg) and x ∈ G is a unitary representation. −

We have 1 ⊂ λG if and only if G is compact.

Examples of Unitary Representations

David Hume Kazhdan’s Property (T) We have 1 ⊂ λG if and only if G is compact.

Examples of Unitary Representations

The trivial representation is deﬁned as 1 ∶ G → C given by 1(g)x = x for all g ∈ G and x ∈ C. A representation π ∶ G → U(H) is said to contain the trivial representation if it has a non-zero invariant vector. This is denoted 1 ⊂ π. Given a locally compact group G with left invariant Haar measure 2 dg, the left regular representation λG ∶ G → U(L (G, dg)) given by 1 2 λG (g)F (x) = F (g x) for all F ∈ L (G, dg) and x ∈ G is a unitary representation. −

David Hume Kazhdan’s Property (T) Examples of Unitary Representations

We have 1 ⊂ λG if and only if G is compact.

David Hume Kazhdan’s Property (T) π has almost invariant vectors if, for every compact K ⊂ G and ε > 0 there is a non-zero (K, ε)-invariant vector vK,ε.

Almost invariant vectors

Let π ∶ G → U(H) be a unitary representation, let K be a subset of G and let ε > 0. A vector v ∈ H is (K, ε)-invariant if

sup Yπ(g)v − vY < εYvY g K

∈

David Hume Kazhdan’s Property (T) Almost invariant vectors

Let π ∶ G → U(H) be a unitary representation, let K be a subset of G and let ε > 0. A vector v ∈ H is (K, ε)-invariant if

sup Yπ(g)v − vY < εYvY g K

∈ π has almost invariant vectors if, for every compact K ⊂ G and ε > 0 there is a non-zero (K, ε)-invariant vector vK,ε.

David Hume Kazhdan’s Property (T) Proof. Fix some compact K ⊂ R and some ε > 0. Choose a ∈ [0, ∞) such 2 2 that K ⊂ [−aε , aε ] and deﬁne v = χ 4a,4a . For each g ∈ K, Sπ(g)v − vY ≤ χB , where 2 2 [−2 ] 2 B = [−a(4 + ε ), −a(4 − ε )] ∪ [a(4 − ε ), a(4 + ε )]. Thus

2 1 1 sup Yπ(g)v − vY ≤ YχB Y = (4aε ) 2 < ε(8a) 2 = εYvY. g K

More generally,∈ a group G is amenable if and only if the left regular representation has almost invariant vectors.

Examples

Lemma 2 λR ∶ R → U(L (R)) has almost invariant vectors.

David Hume Kazhdan’s Property (T) More generally, a group G is amenable if and only if the left regular representation has almost invariant vectors.

Examples

Lemma 2 λR ∶ R → U(L (R)) has almost invariant vectors.

Proof. Fix some compact K ⊂ R and some ε > 0. Choose a ∈ [0, ∞) such 2 2 that K ⊂ [−aε , aε ] and deﬁne v = χ 4a,4a . For each g ∈ K, Sπ(g)v − vY ≤ χB , where 2 2 [−2 ] 2 B = [−a(4 + ε ), −a(4 − ε )] ∪ [a(4 − ε ), a(4 + ε )]. Thus

2 1 1 sup Yπ(g)v − vY ≤ YχB Y = (4aε ) 2 < ε(8a) 2 = εYvY. g K

∈

David Hume Kazhdan’s Property (T) Examples

Lemma 2 λR ∶ R → U(L (R)) has almost invariant vectors.

2 1 1 sup Yπ(g)v − vY ≤ YχB Y = (4aε ) 2 < ε(8a) 2 = εYvY. g K

More generally,∈ a group G is amenable if and only if the left regular representation has almost invariant vectors.

David Hume Kazhdan’s Property (T) Theorem A locally compact amenable group with property (T) is compact.

Proof. We previously established that G is amenable if and only if the left regular representation has almost invariant vectors, and the left regular representation of G contains the trivial representation if and only if G is compact.

Property (T)

A locally compact group G has Property (T) if there exists a compact subset K ⊂ G and some ε > 0 such that every unitary representation with (K, ε)-almost invariant vectors contains the trivial representation (has a non-zero ﬁxed vector).

David Hume Kazhdan’s Property (T) Proof. We previously established that G is amenable if and only if the left regular representation has almost invariant vectors, and the left regular representation of G contains the trivial representation if and only if G is compact.

Property (T)

David Hume Kazhdan’s Property (T) Property (T)

David Hume Kazhdan’s Property (T) Proof. Let Q ⊂ G be compact and ε > 0 be given by the deﬁnition of property (T) for G. Set Q = ψ(G) which is a compact subset of H. ′ Let π ∶ H → U(H) be a unitary representation with a (Q , ε) invariant vector v . Now π ○ ψ ∶ G → U(H) is a unitary ′ representation of G′ and v is (Q, ε) invariant. Hence there is some non-zero vector v which is′ invariant under π(ψ(G)). Since ψ(G) is dense in H and π is strongly continuous, v is invariant under H.

Consequences of Property (T): Quotients

Theorem Let ψ ∶ G → H be a continuous homomorphism between topological groups with dense image. If G has property (T) then H has property (T).

David Hume Kazhdan’s Property (T) Consequences of Property (T): Quotients

Theorem Let ψ ∶ G → H be a continuous homomorphism between topological groups with dense image. If G has property (T) then H has property (T).

Proof. Let Q ⊂ G be compact and ε > 0 be given by the deﬁnition of property (T) for G. Set Q = ψ(G) which is a compact subset of H. ′ Let π ∶ H → U(H) be a unitary representation with a (Q , ε) invariant vector v . Now π ○ ψ ∶ G → U(H) is a unitary ′ representation of G′ and v is (Q, ε) invariant. Hence there is some non-zero vector v which is′ invariant under π(ψ(G)). Since ψ(G) is dense in H and π is strongly continuous, v is invariant under H.

David Hume Kazhdan’s Property (T) Proof. Let C be the set of all open compactly generated subgroups of G. For each H ∈ C, we have that G~H is discrete, so there is a unitary representation

2 1 λG H ∶ G → ` (G~H) given by λG H (x)F (gH) = F (x gH). − ~ ~ Now deﬁne π = >H λG H . Suppose π has a non-zero invariant vector v H vH . Choose H so that vH 0. Now vH is a = > ∈C ~ ≠ non-zero G-invariant vector in `2 G H so must be constant. ∈C ( ~ ) Hence G~H is ﬁnite, and G is compactly generated.

Consequences of Property (T): Compact Generation

Theorem A locally compact group G with Property (T) is compactly generated.

David Hume Kazhdan’s Property (T) Consequences of Property (T): Compact Generation

Theorem A locally compact group G with Property (T) is compactly generated.

Proof. Let C be the set of all open compactly generated subgroups of G. For each H ∈ C, we have that G~H is discrete, so there is a unitary representation

David Hume Kazhdan’s Property (T) Proof. We prove that it has almost invariant vectors. As G has property (T) we deduce that there is an invariant vector. Fix K ⊂ G compact. Since C is an open cover of G, we have K ⊂ H1 ∪ ... ∪ Hn for some Hi ∈ C. Now H = ⟨H1,..., Hn⟩ ∈ C. 2 Deﬁne δH ∈ ` (G~H) to be the Dirac function. We have

Yπ(x)δH − δH Y = 0 for all x ∈ K

2 viewing δH as a vector in >H ` (G~H). ∈C

Consequences of Property (T): Compact Generation

Claim The unitary representation π = >H λG H has a non-zero invariant vector. ∈C ~

David Hume Kazhdan’s Property (T) Consequences of Property (T): Compact Generation

Claim The unitary representation π = >H λG H has a non-zero invariant vector. ∈C ~ Proof. We prove that it has almost invariant vectors. As G has property (T) we deduce that there is an invariant vector. Fix K ⊂ G compact. Since C is an open cover of G, we have K ⊂ H1 ∪ ... ∪ Hn for some Hi ∈ C. Now H = ⟨H1,..., Hn⟩ ∈ C. 2 Deﬁne δH ∈ ` (G~H) to be the Dirac function. We have

Yπ(x)δH − δH Y = 0 for all x ∈ K

2 viewing δH as a vector in >H ` (G~H). ∈C

David Hume Kazhdan’s Property (T) Proof. ab ab The map G → G is continuous and surjective. Thus G has property (T). It is also abelian, and hence amenable. Therefore, G ab is compact.

As a corollary, non-abelian free groups do not have property (T).

Theorem A locally compact group G with property (T) has compact abelianisation.

David Hume Kazhdan’s Property (T) Theorem A locally compact group G with property (T) has compact abelianisation.

Proof. ab ab The map G → G is continuous and surjective. Thus G has property (T). It is also abelian, and hence amenable. Therefore, G ab is compact.

As a corollary, non-abelian free groups do not have property (T).

David Hume Kazhdan’s Property (T) Proof. Let π G be a unitary representation admitting a √∶ → U(H) (G, 2)-invariant vector v. Let C be the closed convex hull of π(G)v. Let v0 be the unique point in C with minimal norm. Since π(g)C = C for all g ∈ G, we have π(g)v0 = v0 for all g ∈ G. √ Set ε = 2 − supg G Yπ(g)v − vY > 0. By the parallelogram law

∈ 2 √ 2 2 − 2Re⟨π(g)v, v⟩ = Yπ(g)v − vY ≤ ( 2 − ε)

ε √ for all g ∈ G. Hence Re⟨π(g)w, w⟩ ≥ 2 (2 2 − ε) > 0 for all w ∈ C. Thus v0 ≠ 0 .

Groups with Property (T): Compact Groups

Theorem Let G be a topological group. If a unitary representation of G √ admits a (G, 2)-invariant vector, then it has an invariant vector. In particular, every compact group G has property (T).

David Hume Kazhdan’s Property (T) Groups with Property (T): Compact Groups

Theorem Let G be a topological group. If a unitary representation of G √ admits a (G, 2)-invariant vector, then it has an invariant vector. In particular, every compact group G has property (T).

Proof. Let π G be a unitary representation admitting a √∶ → U(H) (G, 2)-invariant vector v. Let C be the closed convex hull of π(G)v. Let v0 be the unique point in C with minimal norm. Since π(g)C = C for all g ∈ G, we have π(g)v0 = v0 for all g ∈ G. √ Set ε = 2 − supg G Yπ(g)v − vY > 0. By the parallelogram law

∈ 2 √ 2 2 − 2Re⟨π(g)v, v⟩ = Yπ(g)v − vY ≤ ( 2 − ε)

ε √ for all g ∈ G. Hence Re⟨π(g)w, w⟩ ≥ 2 (2 2 − ε) > 0 for all w ∈ C. Thus v0 ≠ 0 . David Hume Kazhdan’s Property (T) H A group G has bounded generation if there is a ﬁnite subset S ⊂ G and a positive integer N such that for every g ∈ G there is an N ri equality g = Πi 1si where each si ∈ S and ri ∈ Z.

Groups with Property (T): Bounded Generation (Shalom)

2 Starting point: if a unitary representation of SL(2, Z) ⋉ Z has almost invariant vectors, then there is a vector which is invariant 2 2 2 under the action of Z (we say the pair (SL(2, Z) ⋊ Z , Z ) satisﬁes relative Property (T)).

David Hume Kazhdan’s Property (T) Groups with Property (T): Bounded Generation (Shalom)

A group G has bounded generation if there is a ﬁnite subset S ⊂ G and a positive integer N such that for every g ∈ G there is an N ri equality g = Πi 1si where each si ∈ S and ri ∈ Z.

David Hume Kazhdan’s Property (T) Groups with Property (T): Bounded Generation (Shalom)

A group G has bounded generation if there is a ﬁnite subset S ⊂ G and a positive integer N such that for every g ∈ G there is an N ri equality g = Πi 1si where each si ∈ S and ri ∈ Z.

David Hume Kazhdan’s Property (T) Groups with Property (T): Bounded Generation (Shalom)

2 Step 1: Let π ∶ SL2(Z) ⋉ Z → U(H) be a unitary representation. 2 If v is (Q, ~20)-invariant, then it is (Z , )-invariant.

David Hume Kazhdan’s Property (T) Groups with Property (T): Bounded Generation (Shalom)

Step 2: If G is boundedly generated by a set S which can be 2 “covered” by copies of SL2(Z) ⋉ Z , then G has Property (T).

David Hume Kazhdan’s Property (T) There is a natural involution ∗ ∶ RG → RG which extends the map 1 g → g . − The Laplacian∆ ∈ RG is defined by 1 ∆ S s s S 1 s 1 s . = S S − Q = 2 Q ∈ ( − ) ( − ) s S ∗ ∈ A group G satisfies Kazhdan’s Property (T) if there exists some λ > 0 and finitely many elements vi ∈ RG such that

2 ∆ − λ∆ = Q vi vi . i ∗

Groups with Property (T): Group Rings (Ozawa)

Given a group G, generated by a ﬁnite symmetric set S, the group ring RG is the vector space of ﬁnitely support functions G → R.

David Hume Kazhdan’s Property (T) The Laplacian∆ ∈ RG is defined by 1 ∆ S s s S 1 s 1 s . = S S − Q = 2 Q ∈ ( − ) ( − ) s S ∗ ∈ A group G satisfies Kazhdan’s Property (T) if there exists some λ > 0 and finitely many elements vi ∈ RG such that

2 ∆ − λ∆ = Q vi vi . i ∗

Groups with Property (T): Group Rings (Ozawa)

David Hume Kazhdan’s Property (T) A group G satisﬁes Kazhdan’s Property (T) if there exists some λ > 0 and ﬁnitely many elements vi ∈ RG such that

2 ∆ − λ∆ = Q vi vi . i ∗

Groups with Property (T): Group Rings (Ozawa)

Given a group G, generated by a finite symmetric set S, the group ring RG is the vector space of finitely support functions G → R. There is a natural involution ∗ ∶ RG → RG which extends the map 1 g → g . − The Laplacian∆ ∈ RG is defined by 1 ∆ S s s S 1 s 1 s . = S S − Q = 2 Q ∈ ( − ) ( − ) s S ∗ ∈

David Hume Kazhdan’s Property (T) Groups with Property (T): Group Rings (Ozawa)

2 ∆ − λ∆ = Q vi vi . i ∗

David Hume Kazhdan’s Property (T) Groups with Property (T): Group Rings (Ozawa)

David Hume Kazhdan’s Property (T) Questions

David Hume Kazhdan’s Property (T)