RIGIDITY OF GROUP ACTIONS I. Introduction to Super-Rigidity Alex - - PowerPoint PPT Presentation

RIGIDITY OF GROUP ACTIONS

• I. Introduction to Super-Rigidity

Alex Furman (University of Illinois at Chicago) February 28, 2007

The Super-rigidity Phenomenon

For some Γ < G representations ρ : Γ− →H extend to G: Γ

• ρ

H

G

¯ ρ

The Super-rigidity Phenomenon

For some Γ < G representations ρ : Γ− →H extend to G: Γ

• ρ

H

G

¯ ρ

• provided

G is a “higher rank” lcsc group

The Super-rigidity Phenomenon

For some Γ < G representations ρ : Γ− →H extend to G: Γ

• ρ

H

G

¯ ρ

• provided

G is a “higher rank” lcsc group Γ < G – an (irreducible) lattice

The Super-rigidity Phenomenon

For some Γ < G representations ρ : Γ− →H extend to G: Γ

• ρ

H

G

¯ ρ

• provided

G is a “higher rank” lcsc group Γ < G – an (irreducible) lattice ρ : Γ− →H with ρ(Γ) “non-elemntary” in H.

Lattices

Definition

Γ < G is a lattice if Γ is discrete and Haar(G/Γ) < ∞.

Lattices

Definition

Γ < G is a lattice if Γ is discrete and Haar(G/Γ) < ∞. Γ < G = n

i=1 Gi is irreducible if pri(Γ) dense in Gi.

Lattices

Definition

Γ < G is a lattice if Γ is discrete and Haar(G/Γ) < ∞. Γ < G = n

i=1 Gi is irreducible if pri(Γ) dense in Gi.

Examples (Arithmetic)

◮ Γ = Zn in G = Rn

Lattices

Definition

Γ < G is a lattice if Γ is discrete and Haar(G/Γ) < ∞. Γ < G = n

i=1 Gi is irreducible if pri(Γ) dense in Gi.

Examples (Arithmetic)

◮ Γ = Zn in G = Rn ◮ Γ = SLn(Z) in G = SLn(R)

Lattices

Definition

Γ < G is a lattice if Γ is discrete and Haar(G/Γ) < ∞. Γ < G = n

i=1 Gi is irreducible if pri(Γ) dense in Gi.

Examples (Arithmetic)

◮ Γ = Zn in G = Rn ◮ Γ = SLn(Z) in G = SLn(R) ◮ Γ = Z(

√ 2) in G = R2 with (a + b √ 2, a − b √ 2)

Lattices

Definition

Γ < G is a lattice if Γ is discrete and Haar(G/Γ) < ∞. Γ < G = n

i=1 Gi is irreducible if pri(Γ) dense in Gi.

Examples (Arithmetic)

◮ Γ = Zn in G = Rn ◮ Γ = SLn(Z) in G = SLn(R) ◮ Γ = Z(

√ 2) in G = R2 with (a + b √ 2, a − b √ 2)

◮ “similar” construction of Γ in G = SL2(R) × SL2(R)

Lattices

Definition

Γ < G is a lattice if Γ is discrete and Haar(G/Γ) < ∞. Γ < G = n

i=1 Gi is irreducible if pri(Γ) dense in Gi.

Examples (Arithmetic)

◮ Γ = Zn in G = Rn ◮ Γ = SLn(Z) in G = SLn(R) ◮ Γ = Z(

√ 2) in G = R2 with (a + b √ 2, a − b √ 2)

◮ “similar” construction of Γ in G = SL2(R) × SL2(R)

Example (Geometric)

Γ = π1(M) for M – loc. symmetric, compact (or vol(M) < ∞) is a lattice in G = Isom( M).

Margulis’ Higher rank Super-rigidity

Theorem (Superrigidity, Margulis 1970s)

Assume G = Gi – semi-simple Lie group with rk(G) ≥ 2 H – simple and center free Γ < G – an irreducible lattice ρ : Γ− →H with ρ(Γ) Zariski dense in H.

Margulis’ Higher rank Super-rigidity

Theorem (Superrigidity, Margulis 1970s)

Assume G = Gi – semi-simple Lie group with rk(G) ≥ 2 H – simple and center free Γ < G – an irreducible lattice ρ : Γ− →H with ρ(Γ) Zariski dense in H. Then

◮ either ρ(Γ) precompact in H ◮ or ρ : Γ−

→H extends to G

¯ ρ

− →H.

Margulis’ Higher rank Super-rigidity

Theorem (Superrigidity, Margulis 1970s)

Assume G = Gi – semi-simple Lie group with rk(G) ≥ 2 H – simple and center free Γ < G – an irreducible lattice ρ : Γ− →H with ρ(Γ) Zariski dense in H. Then

◮ either ρ(Γ) precompact in H ◮ or ρ : Γ−

→H extends to G

¯ ρ

− →H.

Theorem (Arithmeticity, Margulis 1970s)

In higher rank all irreducible lattices are arithmetic !

Measurable Cocycles

G, H – lcsc groups, G (X, µ) – prob. m.p. action

Measurable Cocycles

G, H – lcsc groups, G (X, µ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀g1, g2 ∈ G : c(g1g2, x) = c(g1, g2.x) · c(g2, x)

Measurable Cocycles

G, H – lcsc groups, G (X, µ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀g1, g2 ∈ G : c(g1g2, x) = c(g1, g2.x) · c(g2, x) Cohomologous cocycles: c ∼ c′ if ∃ f : X → H s.t. c′(g, x) = f (g.x)c(g, x)f (x)−1

Measurable Cocycles

G, H – lcsc groups, G (X, µ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀g1, g2 ∈ G : c(g1g2, x) = c(g1, g2.x) · c(g2, x) Cohomologous cocycles: c ∼ c′ if ∃ f : X → H s.t. c′(g, x) = f (g.x)c(g, x)f (x)−1

Examples

◮ c(g, x) = ρ(g) for some hom ρ : G → H.

Measurable Cocycles

G, H – lcsc groups, G (X, µ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀g1, g2 ∈ G : c(g1g2, x) = c(g1, g2.x) · c(g2, x) Cohomologous cocycles: c ∼ c′ if ∃ f : X → H s.t. c′(g, x) = f (g.x)c(g, x)f (x)−1

Examples

◮ c(g, x) = ρ(g) for some hom ρ : G → H. ◮ σ : G × G/Γ−

→Γ – the “canonical” cocycle

Measurable Cocycles

G, H – lcsc groups, G (X, µ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀g1, g2 ∈ G : c(g1g2, x) = c(g1, g2.x) · c(g2, x) Cohomologous cocycles: c ∼ c′ if ∃ f : X → H s.t. c′(g, x) = f (g.x)c(g, x)f (x)−1

Examples

◮ c(g, x) = ρ(g) for some hom ρ : G → H. ◮ σ : G × G/Γ−

→Γ – the “canonical” cocycle

Observation

{ρ : Γ− →H}/conj ∼ = {c : G × G/Γ → H}/ ∼ .

Zimmer’s Cocycle Super-rigidity

Theorem (Cocycle Super-rigidty, Zimmer 1981)

Let G = Gi be a semi-simple Lie group with rk(G) ≥ 2. G (X, µ) a prob. m.p. action with each Gi ergodic.

Zimmer’s Cocycle Super-rigidity

Theorem (Cocycle Super-rigidty, Zimmer 1981)

Let G = Gi be a semi-simple Lie group with rk(G) ≥ 2. G (X, µ) a prob. m.p. action with each Gi ergodic. H – simple center free, c : G × X → H Zariski dense cocycle.

Zimmer’s Cocycle Super-rigidity

Theorem (Cocycle Super-rigidty, Zimmer 1981)

Let G = Gi be a semi-simple Lie group with rk(G) ≥ 2. G (X, µ) a prob. m.p. action with each Gi ergodic. H – simple center free, c : G × X → H Zariski dense cocycle. Then

◮ either c ∼ c0 : G × X → K with K – compact subgrp in H ◮ or c ∼ ρ : G → H:

c(g, x) = f (g.x)ρ(g)f (x)−1.

Zimmer’s Cocycle Super-rigidity

Theorem (Cocycle Super-rigidty, Zimmer 1981)

Let G = Gi be a semi-simple Lie group with rk(G) ≥ 2. G (X, µ) a prob. m.p. action with each Gi ergodic. H – simple center free, c : G × X → H Zariski dense cocycle. Then

◮ either c ∼ c0 : G × X → K with K – compact subgrp in H ◮ or c ∼ ρ : G → H:

c(g, x) = f (g.x)ρ(g)f (x)−1.

Remarks

◮ Margulis’ super-rigidity corresponds to X = G/Γ

Zimmer’s Cocycle Super-rigidity

Theorem (Cocycle Super-rigidty, Zimmer 1981)

Let G = Gi be a semi-simple Lie group with rk(G) ≥ 2. G (X, µ) a prob. m.p. action with each Gi ergodic. H – simple center free, c : G × X → H Zariski dense cocycle. Then

◮ either c ∼ c0 : G × X → K with K – compact subgrp in H ◮ or c ∼ ρ : G → H:

c(g, x) = f (g.x)ρ(g)f (x)−1.

Remarks

◮ Margulis’ super-rigidity corresponds to X = G/Γ ◮ Proofs combine Algebraic groups with Ergodic Theory

G-boundary (B, ν) = (G/P, Haar) plays a key role

Cocycles: where from and what for ?

◮ Volume preserving Actions on Manifolds

ρ : Γ− →Diff(M, vol)

Cocycles: where from and what for ?

◮ Volume preserving Actions on Manifolds

ρ : Γ− →Diff(M, vol) Γ TM ∼ = Rd × M where d = dim M

• α : Γ × M → GLd(R) or α : Γ × M → SLd(R).

Cocycles: where from and what for ?

◮ Volume preserving Actions on Manifolds

ρ : Γ− →Diff(M, vol) Γ TM ∼ = Rd × M where d = dim M

• α : Γ × M → GLd(R) or α : Γ × M → SLd(R).

◮ Orbit Equivalence in Ergodic Theory

Γ (X, µ) and Λ (Y , ν) free erg. actions

Cocycles: where from and what for ?

◮ Volume preserving Actions on Manifolds

ρ : Γ− →Diff(M, vol) Γ TM ∼ = Rd × M where d = dim M

• α : Γ × M → GLd(R) or α : Γ × M → SLd(R).

◮ Orbit Equivalence in Ergodic Theory

Γ (X, µ) and Λ (Y , ν) free erg. actions OE is T : (X, µ) ∼ = (Y , ν) with T(Γ.x) = Λ.T(x)

Cocycles: where from and what for ?

◮ Volume preserving Actions on Manifolds

ρ : Γ− →Diff(M, vol) Γ TM ∼ = Rd × M where d = dim M

• α : Γ × M → GLd(R) or α : Γ × M → SLd(R).

◮ Orbit Equivalence in Ergodic Theory

Γ (X, µ) and Λ (Y , ν) free erg. actions OE is T : (X, µ) ∼ = (Y , ν) with T(Γ.x) = Λ.T(x)

• αT : Γ × X → Λ by

T(γ.x) = α(g, x).T(x)

Popa’s Cocycle Super-rigidity

Theorem (S.Popa 2006)

Let Γ have (T) and Γ X = (X0, µ0)Γ be a Bernoulli action.

Popa’s Cocycle Super-rigidity

Theorem (S.Popa 2006)

Let Γ have (T) and Γ X = (X0, µ0)Γ be a Bernoulli action. Then for any discrete or compact Λ every cocycle α : Γ × X → Λ is cohomologous to a homomorphism ρ : Γ → Λ.

Popa’s Cocycle Super-rigidity

Theorem (S.Popa 2006)

Let Γ have (T) and Γ X = (X0, µ0)Γ be a Bernoulli action. Then for any discrete or compact Λ every cocycle α : Γ × X → Λ is cohomologous to a homomorphism ρ : Γ → Λ.

Remark

◮ Λ arbitrary discrete or compact (or in Ufin) !

Popa’s Cocycle Super-rigidity

Theorem (S.Popa 2006)

Let Γ have (T) and Γ X = (X0, µ0)Γ be a Bernoulli action. Then for any discrete or compact Λ every cocycle α : Γ × X → Λ is cohomologous to a homomorphism ρ : Γ → Λ.

Remark

◮ Λ arbitrary discrete or compact (or in Ufin) ! ◮ No assumptions on α ! All cocycles ∼ to homs in Λ !

Popa’s Cocycle Super-rigidity

Theorem (S.Popa 2006)

Let Γ have (T) and Γ X = (X0, µ0)Γ be a Bernoulli action. Then for any discrete or compact Λ every cocycle α : Γ × X → Λ is cohomologous to a homomorphism ρ : Γ → Λ.

Remark

◮ Λ arbitrary discrete or compact (or in Ufin) ! ◮ No assumptions on α ! All cocycles ∼ to homs in Λ ! ◮ “deformation-rigidity”: malleability - spectral assumption (T)

Popa’s Cocycle Super-rigidity

Theorem (S.Popa 2006)

Let Γ have (T) and Γ X = (X0, µ0)Γ be a Bernoulli action. Then for any discrete or compact Λ every cocycle α : Γ × X → Λ is cohomologous to a homomorphism ρ : Γ → Λ.

Remark

◮ Λ arbitrary discrete or compact (or in Ufin) ! ◮ No assumptions on α ! All cocycles ∼ to homs in Λ ! ◮ “deformation-rigidity”: malleability - spectral assumption (T) ◮ The assumption on the action Γ X rather than on G or Γ

Popa’s Cocycle Super-rigidity

Theorem (S.Popa 2006)

Let Γ have (T) and Γ X = (X0, µ0)Γ be a Bernoulli action. Then for any discrete or compact Λ every cocycle α : Γ × X → Λ is cohomologous to a homomorphism ρ : Γ → Λ.

Remark

◮ Λ arbitrary discrete or compact (or in Ufin) ! ◮ No assumptions on α ! All cocycles ∼ to homs in Λ ! ◮ “deformation-rigidity”: malleability - spectral assumption (T) ◮ The assumption on the action Γ X rather than on G or Γ ◮ leads to “von Neumann rigidity”

Popa’s Cocycle Super-rigidity

Theorem (S.Popa 2006)

Let Γ have (T) and Γ X = (X0, µ0)Γ be a Bernoulli action. Then for any discrete or compact Λ every cocycle α : Γ × X → Λ is cohomologous to a homomorphism ρ : Γ → Λ.

Remark

◮ Λ arbitrary discrete or compact (or in Ufin) ! ◮ No assumptions on α ! All cocycles ∼ to homs in Λ ! ◮ “deformation-rigidity”: malleability - spectral assumption (T) ◮ The assumption on the action Γ X rather than on G or Γ ◮ leads to “von Neumann rigidity”

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) ????? ????? ????? ????? ????? Γ < G alg Margulis alg G × X Zimmer Γ < Λ < G Margulis Margulis (1974) Zimmer Ann.Math. (1981)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) ????? ????? ????? ????? Γ < G alg Margulis Bu-Mzs alg G × X Zimmer Γ < Λ < G Margulis Bu-Mzs Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp ????? ????? ????? Γ < G alg Margulis Bu-Mzs alg G × X Zimmer Adams Adams Γ < Λ < G Margulis Bu-Mzs Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp ????? ????? ????? Γ < G alg Margulis Bu-Mzs Furst alg G × X Zimmer Adams Adams Γ < Λ < G Margulis Bu-Mzs Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp ????? Modg ????? Γ < G alg Margulis Bu-Mzs Furst K-M alg G × X Zimmer Adams Adams Γ < Λ < G Margulis Bu-Mzs Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp S1 Modg ????? Γ < G alg Margulis Bu-Mzs Furst Ghys K-M alg G × X Zimmer Adams Adams Γ < Λ < G Margulis Bu-Mzs Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996) Ghys Inven. (1999)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp S1 Modg Isom(H) Γ < G alg Margulis Bu-Mzs Furst Ghys K-M (T) alg G × X Zimmer Adams Adams (T) Γ < Λ < G Margulis Bu-Mzs Shalom Γ < Gi Sh Gi × X (Shalom) Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996) Ghys Inven. (1999) Shalom Inven. (2000)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp S1 Modg Isom(H) Γ < G alg Margulis Bu-Mzs Furst Ghys K-M (T) alg G × X Zimmer Adams Adams Wi-Zi (T) Γ < Λ < G Margulis Bu-Mzs Shalom Γ < Gi Sh Gi × X (Shalom) Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996) Ghys Inven. (1999) Shalom Inven. (2000) Witte-Zimmer Geom.Ded.(2001)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp S1 Modg Isom(H) Γ < G alg Margulis Bu-Mzs Furst Ghys K-M (T) alg G × X Zimmer Adams Adams Wi-Zi (T) Γ < Λ < G Margulis Bu-Mzs Shalom Γ < Gi Md-Sh M-M-S Sh Gi × X and H-K and H-K (Shalom) Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996) Ghys Inven. (1999) Shalom Inven. (2000) Witte-Zimmer Geom.Ded.(2001) Monod-Shalom JDG (2004) Mineev-Monod-Shalom Top.(2004) Hjorth-Kechris Mem.AMS (2005)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp S1 Modg Isom(H) Γ < G alg Margulis Bu-Mzs Furst Ghys K-M (T) alg G × X Zimmer Adams Adams Wi-Zi (T) Γ < Λ < G Margulis Bu-Mzs Shalom Γ < Gi Monod Md-Sh M-M-S Sh Md Gi × X and H-K and H-K (Shalom) Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996) Ghys Inven. (1999) Shalom Inven. (2000) Witte-Zimmer Geom.Ded.(2001) Monod-Shalom JDG (2004) Mineev-Monod-Shalom Top.(2004) Hjorth-Kechris Mem.AMS (2005) Monod JAMS (2006)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp S1 Modg Isom(H) Γ < G alg Margulis Bu-Mzs Furst Ghys K-M (T) alg G × X Zimmer Adams Adams Wi-Zi (T) Γ < Λ < G Margulis Bu-Mzs Shalom Γ < Gi Monod Md-Sh M-M-S Sh Md Gi × X (F-Md) and H-K and H-K (Shalom) Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996) Ghys Inven. (1999) Shalom Inven. (2000) Witte-Zimmer Geom.Ded.(2001) Monod-Shalom JDG (2004) Mineev-Monod-Shalom Top.(2004) Hjorth-Kechris Mem.AMS (2005) Monod JAMS (2006) Furman-Monod (2007)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp S1 Modg Isom(H) Γ < G alg Margulis Bu-Mzs Furst Ghys K-M (T) alg G × X Zimmer Adams Adams Wi-Zi (T) Γ < Λ < G Margulis Bu-Mzs B-F-S Shalom Γ < Gi Monod Md-Sh M-M-S B-F-S Sh Md Gi × X (F-Md) and H-K and H-K B-F-S (Shalom) Γ ˜ A2 B-F-S (T) Γ × X B-F-S (T) Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996) Ghys Inven. (1999) Shalom Inven. (2000) Witte-Zimmer Geom.Ded.(2001) Monod-Shalom JDG (2004) Mineev-Monod-Shalom Top.(2004) Hjorth-Kechris Mem.AMS (2005) Monod JAMS (2006) Furman-Monod (2007) Bader-Furman-Shaker (2006)

More Margulis-Zimmer like Super-rigidity results

Targets H H(k) CAT(−1) δ-Hyp S1 Modg Isom(H) Γ < G alg Margulis Bu-Mzs Furst Ghys K-M (T) alg G × X Zimmer Adams Adams Wi-Zi (T) Γ < Λ < G Margulis Bu-Mzs B-F B-F-S Shalom Γ < Gi Monod Md-Sh M-M-S B-F-S Sh Md Gi × X (F-Md) and H-K and H-K B-F-S (Shalom) Γ ˜ A2 B-F B-F B-F-S (T) Γ × X B-F B-F B-F-S (T) Margulis (1974) Zimmer Ann.Math. (1981) Burger-Mozes JAMS (1996) Adams ETDS (1996) Furstenberg Bull.AMS (1967) Kaimanovich-Masur Invent. (1996) Ghys Inven. (1999) Shalom Inven. (2000) Witte-Zimmer Geom.Ded.(2001) Monod-Shalom JDG (2004) Mineev-Monod-Shalom Top.(2004) Hjorth-Kechris Mem.AMS (2005) Monod JAMS (2006) Furman-Monod (2007) Bader-Furman-Shaker (2006) Bader-Furman (2007)

Boundary and the Weyl group

Definition (G-Boundaries, after Burger-Monod)

G – a general lcsc grp. A G-boundary is a msbl G-space (B, [ν]) so that

◮ G (B, [ν]) is amenable ◮ G (B × B, [ν × ν]) erg with Unitary Coefficients

Boundary and the Weyl group

Definition (G-Boundaries, after Burger-Monod)

G – a general lcsc grp. A G-boundary is a msbl G-space (B, [ν]) so that

◮ G (B, [ν]) is amenable ◮ G (B × B, [ν × ν]) erg with Unitary Coefficients

Definition (Weyl Group, (Bader-F, Bader-F-Shaker))

Given a G-boundary (B, ν) let WG,B = Aut (B × B, [ν × ν])G.

Boundary and the Weyl group

Definition (G-Boundaries, after Burger-Monod)

G – a general lcsc grp. A G-boundary is a msbl G-space (B, [ν]) so that

◮ G (B, [ν]) is amenable ◮ G (B × B, [ν × ν]) erg with Unitary Coefficients

Definition (Weyl Group, (Bader-F, Bader-F-Shaker))

Given a G-boundary (B, ν) let WG,B = Aut (B × B, [ν × ν])G.

Examples

◮ G -ss alg, B = G/P then WG,B – the classical Weyl

Boundary and the Weyl group

Definition (G-Boundaries, after Burger-Monod)

G – a general lcsc grp. A G-boundary is a msbl G-space (B, [ν]) so that

◮ G (B, [ν]) is amenable ◮ G (B × B, [ν × ν]) erg with Unitary Coefficients

Definition (Weyl Group, (Bader-F, Bader-F-Shaker))

Given a G-boundary (B, ν) let WG,B = Aut (B × B, [ν × ν])G.

Examples

◮ G -ss alg, B = G/P then WG,B – the classical Weyl

(e.g. G = SLn W = Sn)

Boundary and the Weyl group

Definition (G-Boundaries, after Burger-Monod)

G – a general lcsc grp. A G-boundary is a msbl G-space (B, [ν]) so that

◮ G (B, [ν]) is amenable ◮ G (B × B, [ν × ν]) erg with Unitary Coefficients

Definition (Weyl Group, (Bader-F, Bader-F-Shaker))

Given a G-boundary (B, ν) let WG,B = Aut (B × B, [ν × ν])G.

Examples

◮ G -ss alg, B = G/P then WG,B – the classical Weyl

(e.g. G = SLn W = Sn)

◮ G hyperbolic-like W = Z/2Z

Boundary and the Weyl group

Definition (G-Boundaries, after Burger-Monod)

G – a general lcsc grp. A G-boundary is a msbl G-space (B, [ν]) so that

◮ G (B, [ν]) is amenable ◮ G (B × B, [ν × ν]) erg with Unitary Coefficients

Definition (Weyl Group, (Bader-F, Bader-F-Shaker))

Given a G-boundary (B, ν) let WG,B = Aut (B × B, [ν × ν])G.

Examples

◮ G -ss alg, B = G/P then WG,B – the classical Weyl

(e.g. G = SLn W = Sn)

◮ G hyperbolic-like W = Z/2Z ◮ G amenable, can take trivial B and W

Boundary and the Weyl group

Definition (G-Boundaries, after Burger-Monod)

G – a general lcsc grp. A G-boundary is a msbl G-space (B, [ν]) so that

◮ G (B, [ν]) is amenable ◮ G (B × B, [ν × ν]) erg with Unitary Coefficients

Definition (Weyl Group, (Bader-F, Bader-F-Shaker))

Given a G-boundary (B, ν) let WG,B = Aut (B × B, [ν × ν])G.

Examples

◮ G -ss alg, B = G/P then WG,B – the classical Weyl

(e.g. G = SLn W = Sn)

◮ G hyperbolic-like W = Z/2Z ◮ G amenable, can take trivial B and W ◮ G = n Gi with non-amenble factors, (Z/2Z)n < WG,Q Bi.