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 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. To find such a property, Kazhdan considered unitary representations of groups. Why? 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 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. 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 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. − 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 " > 0 there is a non-zero (K;")-invariant vector vK;". Almost invariant vectors Let π ∶ G → U(H) be a unitary representation, let K be a subset of G and let " > 0. A vector v ∈ H is (K;")-invariant if sup Yπ(g)v − vY < "YvY 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 " > 0. A vector v ∈ H is (K;")-invariant if sup Yπ(g)v − vY < "YvY g K ∈ π has almost invariant vectors if, for every compact K ⊂ G and " > 0 there is a non-zero (K;")-invariant vector vK;". David Hume Kazhdan's Property (T) Proof. Fix some compact K ⊂ R and some " > 0. Choose a ∈ [0; ∞) such 2 2 that K ⊂ [−a" ; a" ] and define v = χ 4a;4a . For each g ∈ K, Sπ(g)v − vY ≤ χB , where 2 2 [−2 ] 2 B = [−a(4 + " ); −a(4 − " )] ∪ [a(4 − " ); a(4 + " )]. Thus 2 1 1 sup Yπ(g)v − vY ≤ YχB Y = (4a" ) 2 < "(8a) 2 = "YvY: 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 " > 0. Choose a ∈ [0; ∞) such 2 2 that K ⊂ [−a" ; a" ] and define v = χ 4a;4a . For each g ∈ K, Sπ(g)v − vY ≤ χB , where 2 2 [−2 ] 2 B = [−a(4 + " ); −a(4 − " )] ∪ [a(4 − " ); a(4 + " )]. Thus 2 1 1 sup Yπ(g)v − vY ≤ YχB Y = (4a" ) 2 < "(8a) 2 = "YvY: g K ∈ David Hume Kazhdan's Property (T) Examples Lemma 2 λR ∶ R → U(L (R)) has almost invariant vectors. Proof. Fix some compact K ⊂ R and some " > 0. Choose a ∈ [0; ∞) such 2 2 that K ⊂ [−a" ; a" ] and define v = χ 4a;4a . For each g ∈ K, Sπ(g)v − vY ≤ χB , where 2 2 [−2 ] 2 B = [−a(4 + " ); −a(4 − " )] ∪ [a(4 − " ); a(4 + " )]. Thus 2 1 1 sup Yπ(g)v − vY ≤ YχB Y = (4a" ) 2 < "(8a) 2 = "YvY: g K More generally,∈ a group G is amenable if and only if the left regular representation has almost invariant vectors.