Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei...
Transcript of Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei...
![Page 1: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/1.jpg)
Finite simple quotients of groups satisfyingproperty (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov
EMS-RSME JOINT MATHEMATICAL WEEKENDBilbao, October 7-9, 2011
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 2: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/2.jpg)
Expanders
Definition: ε-Expander
For 0 < ε ∈ R a graph
X = ( V , E )↑ ↑
vertices edges
is ε-expander if ∀A ⊂ V , with |A| ≤ |V |2 , |∂A| ≥ ε|A|,where ∂A = boundary of A = v ∈ V : dist(v ,A) = 1
Definition: Expander family
A family Xii∈N of k-regular finite graphs is a family of expandersif ∃ε > 0 such that ∀i Xi is an ε-expander and |Xi | → ∞.
Expanders are simultaneously sparse and highly connected.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 3: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/3.jpg)
Expanders
Definition: ε-Expander
For 0 < ε ∈ R a graph
X = ( V , E )↑ ↑
vertices edges
is ε-expander if ∀A ⊂ V , with |A| ≤ |V |2 , |∂A| ≥ ε|A|,where ∂A = boundary of A = v ∈ V : dist(v ,A) = 1
Definition: Expander family
A family Xii∈N of k-regular finite graphs is a family of expandersif ∃ε > 0 such that ∀i Xi is an ε-expander and |Xi | → ∞.
Expanders are simultaneously sparse and highly connected.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 4: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/4.jpg)
Expanders
Definition: ε-Expander
For 0 < ε ∈ R a graph
X = ( V , E )↑ ↑
vertices edges
is ε-expander if ∀A ⊂ V , with |A| ≤ |V |2 , |∂A| ≥ ε|A|,where ∂A = boundary of A = v ∈ V : dist(v ,A) = 1
Definition: Expander family
A family Xii∈N of k-regular finite graphs is a family of expandersif ∃ε > 0 such that ∀i Xi is an ε-expander and |Xi | → ∞.
Expanders are simultaneously sparse and highly connected.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 5: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/5.jpg)
Expanders
Applications of expanders:Network constructionsError-correcting codesCryptographyNumber Theory...
Expander families exist:1973: Pinsker: Using counting arguments
For applications one wants explicit constructions.
1973: Margulis: an explicit construction using property (T ) ofKazhdan
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 6: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/6.jpg)
Expanders
Applications of expanders:Network constructionsError-correcting codesCryptographyNumber Theory...
Expander families exist:1973: Pinsker: Using counting arguments
For applications one wants explicit constructions.
1973: Margulis: an explicit construction using property (T ) ofKazhdan
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 7: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/7.jpg)
Expanders
Applications of expanders:Network constructionsError-correcting codesCryptographyNumber Theory...
Expander families exist:1973: Pinsker: Using counting arguments
For applications one wants explicit constructions.
1973: Margulis: an explicit construction using property (T ) ofKazhdan
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 8: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/8.jpg)
Expanders
Applications of expanders:Network constructionsError-correcting codesCryptographyNumber Theory...
Expander families exist:1973: Pinsker: Using counting arguments
For applications one wants explicit constructions.
1973: Margulis: an explicit construction using property (T ) ofKazhdan
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 9: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/9.jpg)
Property (T )
Definition: Property (T ) (1967, Kazhdan)
Let Γ = 〈S〉 with S finite. Then Γ has property (T ) if ∃ε > 0 suchthat
∀ρ : Γ→ U(H) (where H is a Hilbert space) and ∀v ∈ H⊥0 (whereH0 is the space of Γ-invariant vectors of H)
‖ρ(s)v − v‖ ≥ ε‖v‖,∀s ∈ S .
There are not almost Γ-invariant vectors in H⊥0 .
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 10: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/10.jpg)
Property (T )
Definition: Property (T ) (1967, Kazhdan)
Let Γ = 〈S〉 with S finite. Then Γ has property (T ) if ∃ε > 0 suchthat
∀ρ : Γ→ U(H) (where H is a Hilbert space) and ∀v ∈ H⊥0 (whereH0 is the space of Γ-invariant vectors of H)
‖ρ(s)v − v‖ ≥ ε‖v‖,∀s ∈ S .
There are not almost Γ-invariant vectors in H⊥0 .
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 11: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/11.jpg)
Property (T )
Definition: Property (T ) (1967, Kazhdan)
Let Γ = 〈S〉 with S finite. Then Γ has property (T ) if ∃ε > 0 suchthat
∀ρ : Γ→ U(H) (where H is a Hilbert space) and ∀v ∈ H⊥0 (whereH0 is the space of Γ-invariant vectors of H)
‖ρ(s)v − v‖ ≥ ε‖v‖,∀s ∈ S .
There are not almost Γ-invariant vectors in H⊥0 .
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 12: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/12.jpg)
Construction of expanders using property (T )
Let Γ = 〈S〉 (with S = S−1 finite) is infinite and residually finiteand satisfies property (T )
Let Γii∈N a family of normal subgroups of Γ of finite index suchthat |Γ/Γi | tends to infinity.
Then the Cayley graphs Xi = Cay(Γ/Γi ;S) form a family ofexpanders.
Definition: Property tau (1989 Lubotzky, Zimmer)
Γ has property tau if ∃ε > 0 such that
∀ρ : Γ→ U(H) s.t. ρ(Γ) is finite and ∀v ∈ H⊥0
‖ρ(s)v − v‖ ≥ ε‖v‖, ∀s ∈ S .
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 13: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/13.jpg)
Construction of expanders using property (T )
Let Γ = 〈S〉 (with S = S−1 finite) is infinite and residually finiteand satisfies property (T )
Let Γii∈N a family of normal subgroups of Γ of finite index suchthat |Γ/Γi | tends to infinity.
Then the Cayley graphs Xi = Cay(Γ/Γi ;S) form a family ofexpanders.
Definition: Property tau (1989 Lubotzky, Zimmer)
Γ has property tau if ∃ε > 0 such that
∀ρ : Γ→ U(H) s.t. ρ(Γ) is finite and ∀v ∈ H⊥0
‖ρ(s)v − v‖ ≥ ε‖v‖, ∀s ∈ S .
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 14: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/14.jpg)
Construction of expanders using property (T )
Let Γ = 〈S〉 (with S = S−1 finite) is infinite and residually finiteand satisfies property (T )
Let Γii∈N a family of normal subgroups of Γ of finite index suchthat |Γ/Γi | tends to infinity.
Then the Cayley graphs Xi = Cay(Γ/Γi ;S) form a family ofexpanders.
Definition: Property tau (1989 Lubotzky, Zimmer)
Γ has property tau if ∃ε > 0 such that
∀ρ : Γ→ U(H) s.t. ρ(Γ) is finite and ∀v ∈ H⊥0
‖ρ(s)v − v‖ ≥ ε‖v‖, ∀s ∈ S .
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 15: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/15.jpg)
Construction of expanders using property (T )
Let Γ = 〈S〉 (with S = S−1 finite) is infinite and residually finiteand satisfies property (T )
Let Γii∈N a family of normal subgroups of Γ of finite index suchthat |Γ/Γi | tends to infinity.
Then the Cayley graphs Xi = Cay(Γ/Γi ;S) form a family ofexpanders.
Definition: Property tau (1989 Lubotzky, Zimmer)
Γ has property tau if ∃ε > 0 such that
∀ρ : Γ→ U(H) s.t. ρ(Γ) is finite and ∀v ∈ H⊥0
‖ρ(s)v − v‖ ≥ ε‖v‖, ∀s ∈ S .
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 16: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/16.jpg)
Construction of expanders using property (T )
Let Γ = 〈S〉 (with S = S−1 finite) is infinite and residually finiteand satisfies property (T )
Let Γii∈N a family of normal subgroups of Γ of finite index suchthat |Γ/Γi | tends to infinity.
Then the Cayley graphs Xi = Cay(Γ/Γi ;S) form a family ofexpanders.
Definition: Property tau (1989 Lubotzky, Zimmer)
Γ has property tau if ∃ε > 0 such that
∀ρ : Γ→ U(H) s.t. ρ(Γ) is finite and ∀v ∈ H⊥0
‖ρ(s)v − v‖ ≥ ε‖v‖, ∀s ∈ S .
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 17: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/17.jpg)
Family of group expanders
Definition: family of group expanders
A family Gi of finite groups is a family of expanders if ∃k ∈ Nand ε > 0 such that every group Gi has a symmetric subset Si of kgenerators for which Cay(Gi ; Si ) is an ε-expander.
Example: If Γ is a finitely generated group satisfying property (T )or tau, then
finite quotients of Γ
is a family of expanders.
Conjecture (1989 Babai, Kantor and Lubotzky)
The family of all the finite (nonabelian) simple groups is a familyof group expanders.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 18: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/18.jpg)
Family of group expanders
Definition: family of group expanders
A family Gi of finite groups is a family of expanders if ∃k ∈ Nand ε > 0 such that every group Gi has a symmetric subset Si of kgenerators for which Cay(Gi ; Si ) is an ε-expander.
Example: If Γ is a finitely generated group satisfying property (T )or tau, then
finite quotients of Γ
is a family of expanders.
Conjecture (1989 Babai, Kantor and Lubotzky)
The family of all the finite (nonabelian) simple groups is a familyof group expanders.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 19: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/19.jpg)
Family of group expanders
Definition: family of group expanders
A family Gi of finite groups is a family of expanders if ∃k ∈ Nand ε > 0 such that every group Gi has a symmetric subset Si of kgenerators for which Cay(Gi ; Si ) is an ε-expander.
Example: If Γ is a finitely generated group satisfying property (T )or tau, then
finite quotients of Γ
is a family of expanders.
Conjecture (1989 Babai, Kantor and Lubotzky)
The family of all the finite (nonabelian) simple groups is a familyof group expanders.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 20: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/20.jpg)
Family of group expanders
Theorem (Kassabov, Nikolov, Lubotzky, Breulliard, Green, Tao)
There exists k ∈ N and ε > 0 such that every every non-abelianfinite simple group G has a set S of k generators for whichCay(G ; S) is an ε-expander.
2005 M. Kassabov: PSLn(Fq)(n ≥ 3) and Altn2005 N. Nikolov: classical groups of large rank2005 A. Lubotzky: PSL2(Fq) and simple groups of Lie type ofbounded rank with the exception of the Suzuki groups2010 E. Breulliard, B. Green and T. Tao: Suzuki groups.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 21: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/21.jpg)
Family of group expanders
Theorem (Kassabov, Nikolov, Lubotzky, Breulliard, Green, Tao)
There exists k ∈ N and ε > 0 such that every every non-abelianfinite simple group G has a set S of k generators for whichCay(G ; S) is an ε-expander.
2005 M. Kassabov: PSLn(Fq)(n ≥ 3) and Altn2005 N. Nikolov: classical groups of large rank2005 A. Lubotzky: PSL2(Fq) and simple groups of Lie type ofbounded rank with the exception of the Suzuki groups2010 E. Breulliard, B. Green and T. Tao: Suzuki groups.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 22: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/22.jpg)
Family of group expanders
Theorem (Kassabov, Nikolov, Lubotzky, Breulliard, Green, Tao)
There exists k ∈ N and ε > 0 such that every every non-abelianfinite simple group G has a set S of k generators for whichCay(G ; S) is an ε-expander.
2005 M. Kassabov: PSLn(Fq)(n ≥ 3) and Altn2005 N. Nikolov: classical groups of large rank2005 A. Lubotzky: PSL2(Fq) and simple groups of Lie type ofbounded rank with the exception of the Suzuki groups2010 E. Breulliard, B. Green and T. Tao: Suzuki groups.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 23: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/23.jpg)
Family of group expanders
Theorem (Kassabov, Nikolov, Lubotzky, Breulliard, Green, Tao)
There exists k ∈ N and ε > 0 such that every every non-abelianfinite simple group G has a set S of k generators for whichCay(G ; S) is an ε-expander.
2005 M. Kassabov: PSLn(Fq)(n ≥ 3) and Altn2005 N. Nikolov: classical groups of large rank2005 A. Lubotzky: PSL2(Fq) and simple groups of Lie type ofbounded rank with the exception of the Suzuki groups2010 E. Breulliard, B. Green and T. Tao: Suzuki groups.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 24: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/24.jpg)
Family of group expanders
Theorem (Kassabov, Nikolov, Lubotzky, Breulliard, Green, Tao)
There exists k ∈ N and ε > 0 such that every every non-abelianfinite simple group G has a set S of k generators for whichCay(G ; S) is an ε-expander.
2005 M. Kassabov: PSLn(Fq)(n ≥ 3) and Altn2005 N. Nikolov: classical groups of large rank2005 A. Lubotzky: PSL2(Fq) and simple groups of Lie type ofbounded rank with the exception of the Suzuki groups2010 E. Breulliard, B. Green and T. Tao: Suzuki groups.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 25: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/25.jpg)
Mother group
Definition: mother group
Let F be a family of groups. We say that a group Γ is a mothergroup for F if Γ maps onto every group from F .
Question.
Let F be a family of finite simple groups. Does F have a finitelygenerated mother group satisfying Kazhdan’s property (T ) orproperty tau?
Example: SL3(Z) is a mother group of PSL3(Fp).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 26: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/26.jpg)
Mother group
Definition: mother group
Let F be a family of groups. We say that a group Γ is a mothergroup for F if Γ maps onto every group from F .
Question.
Let F be a family of finite simple groups. Does F have a finitelygenerated mother group satisfying Kazhdan’s property (T ) orproperty tau?
Example: SL3(Z) is a mother group of PSL3(Fp).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 27: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/27.jpg)
Mother group
Definition: mother group
Let F be a family of groups. We say that a group Γ is a mothergroup for F if Γ maps onto every group from F .
Question.
Let F be a family of finite simple groups. Does F have a finitelygenerated mother group satisfying Kazhdan’s property (T ) orproperty tau?
Example: SL3(Z) is a mother group of PSL3(Fp).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 28: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/28.jpg)
Mother group
Conjecture
1 The family of all the non-abelian finite simple groups has afinitely generated mother group having property tau.
2 A family F of non-abelian finite simple groups has a mothergroup having property (T ) if and only if only finitely manyfinite simple groups of Lie type of rank 1 belongs to F .
Why do we exclude the groups of Lie type of rank 1 in (2)?
Theorem (folklore)
Let Γ maps on infinitely many PSL2(Fq). Then Γ does not haveproperty (T ).
SL2(Z[1/2]) has property tau and maps onto PSL2(Fp)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 29: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/29.jpg)
Mother group
Conjecture
1 The family of all the non-abelian finite simple groups has afinitely generated mother group having property tau.
2 A family F of non-abelian finite simple groups has a mothergroup having property (T ) if and only if only finitely manyfinite simple groups of Lie type of rank 1 belongs to F .
Why do we exclude the groups of Lie type of rank 1 in (2)?
Theorem (folklore)
Let Γ maps on infinitely many PSL2(Fq). Then Γ does not haveproperty (T ).
SL2(Z[1/2]) has property tau and maps onto PSL2(Fp)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 30: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/30.jpg)
Mother group
Conjecture
1 The family of all the non-abelian finite simple groups has afinitely generated mother group having property tau.
2 A family F of non-abelian finite simple groups has a mothergroup having property (T ) if and only if only finitely manyfinite simple groups of Lie type of rank 1 belongs to F .
Why do we exclude the groups of Lie type of rank 1 in (2)?
Theorem (folklore)
Let Γ maps on infinitely many PSL2(Fq). Then Γ does not haveproperty (T ).
SL2(Z[1/2]) has property tau and maps onto PSL2(Fp)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 31: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/31.jpg)
Mother group
Conjecture
1 The family of all the non-abelian finite simple groups has afinitely generated mother group having property tau.
2 A family F of non-abelian finite simple groups has a mothergroup having property (T ) if and only if only finitely manyfinite simple groups of Lie type of rank 1 belongs to F .
Why do we exclude the groups of Lie type of rank 1 in (2)?
Theorem (folklore)
Let Γ maps on infinitely many PSL2(Fq). Then Γ does not haveproperty (T ).
SL2(Z[1/2]) has property tau and maps onto PSL2(Fp)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 32: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/32.jpg)
Main result
Theorem
There exists a group Γ satisfying property (T ) such that everyfinite simple group of Lie type of rank at least 2 is a quotient of Γ.
Main tool:A method that allows to prove that some groups graded by rootsystems have property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 33: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/33.jpg)
Main result
Theorem
There exists a group Γ satisfying property (T ) such that everyfinite simple group of Lie type of rank at least 2 is a quotient of Γ.
Main tool:A method that allows to prove that some groups graded by rootsystems have property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 34: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/34.jpg)
Root systems
Definition
Let E be real vector space. A finite non-empty subset Φ of E iscalled a root system in E if
(a) Φ spans E ;
(b) Φ does not contain 0;
(c) Φ is closed under inversion, that is, if α ∈ Φ then −α ∈ Φ.
The dimension of E is called the rank of Φ.
Definition
A root system Φ in a space E will be called classical if E can begiven the structure of a Euclidean space such that
(a) For any α, β ∈ Φ we have 2(α,β)(β,β) ∈ Z;
(b) If α, β ∈ Φ, then α− 2(α,β)(β,β) β ∈ Φ
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 35: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/35.jpg)
Root systems
Definition
Let E be real vector space. A finite non-empty subset Φ of E iscalled a root system in E if
(a) Φ spans E ;
(b) Φ does not contain 0;
(c) Φ is closed under inversion, that is, if α ∈ Φ then −α ∈ Φ.
The dimension of E is called the rank of Φ.
Definition
A root system Φ in a space E will be called classical if E can begiven the structure of a Euclidean space such that
(a) For any α, β ∈ Φ we have 2(α,β)(β,β) ∈ Z;
(b) If α, β ∈ Φ, then α− 2(α,β)(β,β) β ∈ Φ
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 36: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/36.jpg)
Root systems
Definition
Let E be real vector space. A finite non-empty subset Φ of E iscalled a root system in E if
(a) Φ spans E ;
(b) Φ does not contain 0;
(c) Φ is closed under inversion, that is, if α ∈ Φ then −α ∈ Φ.
The dimension of E is called the rank of Φ.
Definition
A root system Φ in a space E will be called classical if E can begiven the structure of a Euclidean space such that
(a) For any α, β ∈ Φ we have 2(α,β)(β,β) ∈ Z;
(b) If α, β ∈ Φ, then α− 2(α,β)(β,β) β ∈ Φ
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 37: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/37.jpg)
Root systems
Remark: Every irreducible classical root system is isomorphicto one of the following: An,Bn(n ≥ 2),Cn(n ≥ 3),BCn(n ≥1),Dn(n ≥ 4),E6,E7,E8,F4,G2.
α
J • I
IJ
β
J
I
A2
α
J •
N
I
II
JJ
β
J J
I I
H
G2
α
J •N
I
IJ
β
J HI
B2
α
N
J J •N
I
IJ
β I
J HI
H
BC2
Figure: Classical irreducible root systems of rank 2.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 38: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/38.jpg)
Root systems
Remark: Every irreducible classical root system is isomorphicto one of the following: An,Bn(n ≥ 2),Cn(n ≥ 3),BCn(n ≥1),Dn(n ≥ 4),E6,E7,E8,F4,G2.
α
J • I
IJ
β
J
I
A2
α
J •
N
I
II
JJ
β
J J
I I
H
G2
α
J •N
I
IJ
β
J HI
B2
α
N
J J •N
I
IJ
β I
J HI
H
BC2
Figure: Classical irreducible root systems of rank 2.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 39: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/39.jpg)
Groups graded by root systems
Definition: Φ-grading
Φ a root system, G a group. A Φ-grading of G is a collection ofsubgroups Xαα∈Φ of G , called root subgroups such that
(i) Xαα∈Φ generate G
(ii) For any α, β ∈ Φ, with α 6∈ R<0β, we have
[Xα,Xβ] ⊆ 〈Xγ | γ = aα + bβ ∈ Φ, a, b ≥ 1〉
Informal definition: graded cover
A graded cover of a grading Xαα∈Φ is the quotient of the freeproduct of Xαα∈Φ by the normal subgroup generated byrelations that appear in (ii) of the previous definition
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 40: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/40.jpg)
Groups graded by root systems
Definition: Φ-grading
Φ a root system, G a group. A Φ-grading of G is a collection ofsubgroups Xαα∈Φ of G , called root subgroups such that
(i) Xαα∈Φ generate G
(ii) For any α, β ∈ Φ, with α 6∈ R<0β, we have
[Xα,Xβ] ⊆ 〈Xγ | γ = aα + bβ ∈ Φ, a, b ≥ 1〉
Informal definition: graded cover
A graded cover of a grading Xαα∈Φ is the quotient of the freeproduct of Xαα∈Φ by the normal subgroup generated byrelations that appear in (ii) of the previous definition
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 41: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/41.jpg)
Groups graded by root systems
Theorem (2010, Ershov, Jaikin, Kassabov)
Let Φ be a regular (for example, classical) root system of rank atleast two. Let G be a finitely generated group and Xαα∈Φ itsΦ-grading. Assume that
1 the grading Xα is strong and
2 the pair (G ,∪α∈ΦXα) has relative Kazhdan property,
then G has property (T ).
2008 Ershov, Jaikin: the case of groups with A2-grading
Theorem(2008, Ershov, Jaikin)
Let R be a finitely generated ring and n ≥ 3. Let G = ELn(R),that is, the subgroup of GLn(R) generated by elementary matrices.Then G has property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 42: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/42.jpg)
Groups graded by root systems
Theorem (2010, Ershov, Jaikin, Kassabov)
Let Φ be a regular (for example, classical) root system of rank atleast two. Let G be a finitely generated group and Xαα∈Φ itsΦ-grading. Assume that
1 the grading Xα is strong and
2 the pair (G ,∪α∈ΦXα) has relative Kazhdan property,
then G has property (T ).
2008 Ershov, Jaikin: the case of groups with A2-grading
Theorem(2008, Ershov, Jaikin)
Let R be a finitely generated ring and n ≥ 3. Let G = ELn(R),that is, the subgroup of GLn(R) generated by elementary matrices.Then G has property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 43: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/43.jpg)
Groups graded by root systems
Theorem (2010, Ershov, Jaikin, Kassabov)
Let Φ be a regular (for example, classical) root system of rank atleast two. Let G be a finitely generated group and Xαα∈Φ itsΦ-grading. Assume that
1 the grading Xα is strong and
2 the pair (G ,∪α∈ΦXα) has relative Kazhdan property,
then G has property (T ).
2008 Ershov, Jaikin: the case of groups with A2-grading
Theorem(2008, Ershov, Jaikin)
Let R be a finitely generated ring and n ≥ 3. Let G = ELn(R),that is, the subgroup of GLn(R) generated by elementary matrices.Then G has property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 44: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/44.jpg)
Groups graded by root systems
The simple groups of Lie type have natural gradingsType of simple group Classical group Graded cover
An(q) PSLn+1(Fq) StAn(Fq)
Bn(q) (q is odd) Ω2n+1(Fq) StBn(Fq)
Cn(q) (q 6= 2 if n = 2) PSp2n(Fq) StCn(Fq)
Dn(q) PΩ+2n(Fq) StDn(Fq)
Φ(q) (Φ = G2,En StΦ(Fq)or F4 and q 6= 2 if Φ = G2)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 45: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/45.jpg)
Groups graded by root systems
The simple groups of Lie type have natural gradingsType of simple group Classical group Graded cover
An(q) PSLn+1(Fq) StAn(Fq)
Bn(q) (q is odd) Ω2n+1(Fq) StBn(Fq)
Cn(q) (q 6= 2 if n = 2) PSp2n(Fq) StCn(Fq)
Dn(q) PΩ+2n(Fq) StDn(Fq)
Φ(q) (Φ = G2,En StΦ(Fq)or F4 and q 6= 2 if Φ = G2)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 46: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/46.jpg)
Groups graded by root systems
Type of simple group Classical group Graded cover
2A2m−1(q) PSU2m(Fq) St1Cm
(Fq2 , ∗),
2A2m(q) PSU2m+1(Fq) StBCm(Fq2 , ∗),
2Dm(q) PΩ−2m(Fq) St1Bm−1
(Fq2 , Id , σ)
3D4(q) StG2(Fq3 , σ),
2E6(q) StF4(Fq2 , σ),
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 47: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/47.jpg)
Groups graded by root systems
2F4(2k)(k ≥ 1) is graded and its graded cover is St2F4(F22k+1)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 48: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/48.jpg)
Groups graded by root systems
2F4(2k)(k ≥ 1) is graded and its graded cover is St2F4(F22k+1)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 49: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/49.jpg)
Main result
Theorem
There exists a group Γ satisfying property (T ) such that everyfinite simple group of Lie type of rank at least 2 is a quotient of Γ.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 50: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/50.jpg)
Bounded rank case
Bounded rank case
For each r , there exists a group Γr satisfying property (T ) suchthat every finite simple group of Lie type of rank at least 2 and atmost r is a quotient of Γr .
Proof in the non-twisted case:
We fix a root system Φ.
StΦ(Fq) is the graded cover of the simple group Φ(q) (with a finitenumber of exceptions).
StΦ(Z[t]) maps onto StΦ(Fq).
StΦ(Z[t]) has property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 51: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/51.jpg)
Bounded rank case
Bounded rank case
For each r , there exists a group Γr satisfying property (T ) suchthat every finite simple group of Lie type of rank at least 2 and atmost r is a quotient of Γr .
Proof in the non-twisted case:
We fix a root system Φ.
StΦ(Fq) is the graded cover of the simple group Φ(q) (with a finitenumber of exceptions).
StΦ(Z[t]) maps onto StΦ(Fq).
StΦ(Z[t]) has property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 52: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/52.jpg)
Bounded rank case
Bounded rank case
For each r , there exists a group Γr satisfying property (T ) suchthat every finite simple group of Lie type of rank at least 2 and atmost r is a quotient of Γr .
Proof in the non-twisted case:
We fix a root system Φ.
StΦ(Fq) is the graded cover of the simple group Φ(q) (with a finitenumber of exceptions).
StΦ(Z[t]) maps onto StΦ(Fq).
StΦ(Z[t]) has property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 53: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/53.jpg)
Bounded rank case
Bounded rank case
For each r , there exists a group Γr satisfying property (T ) suchthat every finite simple group of Lie type of rank at least 2 and atmost r is a quotient of Γr .
Proof in the non-twisted case:
We fix a root system Φ.
StΦ(Fq) is the graded cover of the simple group Φ(q) (with a finitenumber of exceptions).
StΦ(Z[t]) maps onto StΦ(Fq).
StΦ(Z[t]) has property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 54: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/54.jpg)
Bounded rank case
Bounded rank case
For each r , there exists a group Γr satisfying property (T ) suchthat every finite simple group of Lie type of rank at least 2 and atmost r is a quotient of Γr .
Proof in the non-twisted case:
We fix a root system Φ.
StΦ(Fq) is the graded cover of the simple group Φ(q) (with a finitenumber of exceptions).
StΦ(Z[t]) maps onto StΦ(Fq).
StΦ(Z[t]) has property (T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 55: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/55.jpg)
Mother group with property (T) for PSLn(Fq)A2 = ±α,±β,±(α + β)A2-grading of SLn(Fq), with n = k + l + m:
Xα =
1 Matk×l(Fq) 00 1 00 0 1
,Xβ =
1 0 00 1 Matl×m(Fq)0 0 1
Xα+β =
1 0 Matk×m(Fq)0 1 00 0 1
,X−γ = (Xγ)transp
EL3(Matk×k(Fq)) ∼= SL3k(Fq)
Let R = Z < x , y >, then EL3(R) maps onto EL3(Matk×k(Fq).
This proves that PSL3k(Fq) has a mother group with property(T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 56: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/56.jpg)
Mother group with property (T) for PSLn(Fq)A2 = ±α,±β,±(α + β)A2-grading of SLn(Fq), with n = k + l + m:
Xα =
1 Matk×l(Fq) 00 1 00 0 1
,Xβ =
1 0 00 1 Matl×m(Fq)0 0 1
Xα+β =
1 0 Matk×m(Fq)0 1 00 0 1
,X−γ = (Xγ)transp
EL3(Matk×k(Fq)) ∼= SL3k(Fq)
Let R = Z < x , y >, then EL3(R) maps onto EL3(Matk×k(Fq).
This proves that PSL3k(Fq) has a mother group with property(T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 57: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/57.jpg)
Mother group with property (T) for PSLn(Fq)A2 = ±α,±β,±(α + β)A2-grading of SLn(Fq), with n = k + l + m:
Xα =
1 Matk×l(Fq) 00 1 00 0 1
,Xβ =
1 0 00 1 Matl×m(Fq)0 0 1
Xα+β =
1 0 Matk×m(Fq)0 1 00 0 1
,X−γ = (Xγ)transp
EL3(Matk×k(Fq)) ∼= SL3k(Fq)
Let R = Z < x , y >, then EL3(R) maps onto EL3(Matk×k(Fq).
This proves that PSL3k(Fq) has a mother group with property(T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 58: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/58.jpg)
Mother group with property (T) for PSLn(Fq)A2 = ±α,±β,±(α + β)A2-grading of SLn(Fq), with n = k + l + m:
Xα =
1 Matk×l(Fq) 00 1 00 0 1
,Xβ =
1 0 00 1 Matl×m(Fq)0 0 1
Xα+β =
1 0 Matk×m(Fq)0 1 00 0 1
,X−γ = (Xγ)transp
EL3(Matk×k(Fq)) ∼= SL3k(Fq)
Let R = Z < x , y >, then EL3(R) maps onto EL3(Matk×k(Fq).
This proves that PSL3k(Fq) has a mother group with property(T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 59: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/59.jpg)
Mother group with property (T) for PSLn(Fq)A2 = ±α,±β,±(α + β)A2-grading of SLn(Fq), with n = k + l + m:
Xα =
1 Matk×l(Fq) 00 1 00 0 1
,Xβ =
1 0 00 1 Matl×m(Fq)0 0 1
Xα+β =
1 0 Matk×m(Fq)0 1 00 0 1
,X−γ = (Xγ)transp
EL3(Matk×k(Fq)) ∼= SL3k(Fq)
Let R = Z < x , y >, then EL3(R) maps onto EL3(Matk×k(Fq).
This proves that PSL3k(Fq) has a mother group with property(T ).
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 60: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/60.jpg)
Mother group with property (T) for symplectic groups
Let J =
0 0 0 Ik0 0 Ik 00 −Ik 0 0−Ik 0 0 0
∈ Mat4k(Fq)
Sp4k(Fq) = A ∈ GL4k(Fq) : AtranspJA = J
There is no a symplectic group over non-commutative ring.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 61: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/61.jpg)
Mother group with property (T) for symplectic groups
Let J =
0 0 0 Ik0 0 Ik 00 −Ik 0 0−Ik 0 0 0
∈ Mat4k(Fq)
Sp4k(Fq) = A ∈ GL4k(Fq) : AtranspJA = J
There is no a symplectic group over non-commutative ring.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 62: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/62.jpg)
Mother group with property (T) for symplectic groups
Let J =
0 0 0 Ik0 0 Ik 00 −Ik 0 0−Ik 0 0 0
∈ Mat4k(Fq)
Sp4k(Fq) = A ∈ GL4k(Fq) : AtranspJA = J
There is no a symplectic group over non-commutative ring.
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 63: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/63.jpg)
Mother group with property (T) for symplectic groups
C2-grading of Sp4k(Fq) (A,B = Btransp ∈ Matk(Fq))
Xe1−e2 =
1 A 0 00 1 0 00 0 1 −tA0 0 0 1
, Xe1+e2 =
1 0 A 00 1 0 tA0 0 1 00 0 0 1
,
X2e1 =
1 0 0 B0 1 0 00 0 1 00 0 0 1
, X2e2 =
1 0 0 00 1 B 00 0 1 00 0 0 1
X−γ = (Xγ)transp
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 64: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/64.jpg)
Mother group with property (T) for symplectic groups
Let R = Z < x , y > be a free ring and ∗ the involution of R suchthat x = x∗ and y = y∗. Let G be a subgroup of EL4(R)generated by the following root subgroups (a, b = b∗ ∈ R)
Xe1−e2 =
1 a 0 00 1 0 00 0 1 −a∗0 0 0 1
, Xe1+e2 =
1 0 a 00 1 0 a∗
0 0 1 00 0 0 1
,
X2e1 =
1 0 0 b0 1 0 00 0 1 00 0 0 1
, X2e2 =
1 0 0 00 1 b 00 0 1 00 0 0 1
,X−γ = (Xγ)transp
Then G has property (T) (we prove it using our method) andG maps onto Sp4k(Fq) (Matk(Fq) can be generated as a ring bytwo symmetric matrices)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 65: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/65.jpg)
Mother group with property (T) for symplectic groups
Let R = Z < x , y > be a free ring and ∗ the involution of R suchthat x = x∗ and y = y∗. Let G be a subgroup of EL4(R)generated by the following root subgroups (a, b = b∗ ∈ R)
Xe1−e2 =
1 a 0 00 1 0 00 0 1 −a∗0 0 0 1
, Xe1+e2 =
1 0 a 00 1 0 a∗
0 0 1 00 0 0 1
,
X2e1 =
1 0 0 b0 1 0 00 0 1 00 0 0 1
, X2e2 =
1 0 0 00 1 b 00 0 1 00 0 0 1
,X−γ = (Xγ)transp
Then G has property (T) (we prove it using our method) andG maps onto Sp4k(Fq) (Matk(Fq) can be generated as a ring bytwo symmetric matrices)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 66: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/66.jpg)
Mother group with property (T) for symplectic groups
Let R = Z < x , y > be a free ring and ∗ the involution of R suchthat x = x∗ and y = y∗. Let G be a subgroup of EL4(R)generated by the following root subgroups (a, b = b∗ ∈ R)
Xe1−e2 =
1 a 0 00 1 0 00 0 1 −a∗0 0 0 1
, Xe1+e2 =
1 0 a 00 1 0 a∗
0 0 1 00 0 0 1
,
X2e1 =
1 0 0 b0 1 0 00 0 1 00 0 0 1
, X2e2 =
1 0 0 00 1 b 00 0 1 00 0 0 1
,X−γ = (Xγ)transp
Then G has property (T) (we prove it using our method) andG maps onto Sp4k(Fq) (Matk(Fq) can be generated as a ring bytwo symmetric matrices)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 67: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/67.jpg)
Mother group with property (T) for symplectic groups
Let R = Z < x , y > be a free ring and ∗ the involution of R suchthat x = x∗ and y = y∗. Let G be a subgroup of EL4(R)generated by the following root subgroups (a, b = b∗ ∈ R)
Xe1−e2 =
1 a 0 00 1 0 00 0 1 −a∗0 0 0 1
, Xe1+e2 =
1 0 a 00 1 0 a∗
0 0 1 00 0 0 1
,
X2e1 =
1 0 0 b0 1 0 00 0 1 00 0 0 1
, X2e2 =
1 0 0 00 1 b 00 0 1 00 0 0 1
,X−γ = (Xγ)transp
Then G has property (T) (we prove it using our method) andG maps onto Sp4k(Fq) (Matk(Fq) can be generated as a ring bytwo symmetric matrices)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 68: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/68.jpg)
Mother group with property (T) for symplectic groups
Let R = Z < x , y > be a free ring and ∗ the involution of R suchthat x = x∗ and y = y∗. Let G be a subgroup of EL4(R)generated by the following root subgroups (a, b = b∗ ∈ R)
Xe1−e2 =
1 a 0 00 1 0 00 0 1 −a∗0 0 0 1
, Xe1+e2 =
1 0 a 00 1 0 a∗
0 0 1 00 0 0 1
,
X2e1 =
1 0 0 b0 1 0 00 0 1 00 0 0 1
, X2e2 =
1 0 0 00 1 b 00 0 1 00 0 0 1
,X−γ = (Xγ)transp
Then G has property (T) (we prove it using our method) andG maps onto Sp4k(Fq) (Matk(Fq) can be generated as a ring bytwo symmetric matrices)
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 69: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/69.jpg)
Open questions
Question 1
Assume that G maps on infinitely many twisted Lie groups of rank1:
2A2(q) = PSU3(Fq), Suzuki groups 2B2(22n+1)
or Ree groups 2G2(32n+1).
Is it true that G can not have property (T )?
Question 2
Is there a finitely generated group with property tau that maps onall PSL2(Fq)?
Question 3
Is there a finitely generated group with property (T ) or tau thatmaps on infinitely many (all) Altn?
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 70: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/70.jpg)
Open questions
Question 1
Assume that G maps on infinitely many twisted Lie groups of rank1:
2A2(q) = PSU3(Fq), Suzuki groups 2B2(22n+1)
or Ree groups 2G2(32n+1).
Is it true that G can not have property (T )?
Question 2
Is there a finitely generated group with property tau that maps onall PSL2(Fq)?
Question 3
Is there a finitely generated group with property (T ) or tau thatmaps on infinitely many (all) Altn?
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 71: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/71.jpg)
Open questions
Question 1
Assume that G maps on infinitely many twisted Lie groups of rank1:
2A2(q) = PSU3(Fq), Suzuki groups 2B2(22n+1)
or Ree groups 2G2(32n+1).
Is it true that G can not have property (T )?
Question 2
Is there a finitely generated group with property tau that maps onall PSL2(Fq)?
Question 3
Is there a finitely generated group with property (T ) or tau thatmaps on infinitely many (all) Altn?
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).
![Page 72: Finite simple quotients of groups satisfying property (T). · 2011-10-26 · Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov EMS-RSME JOINT MATHEMATICAL WEEKEND Bilbao,](https://reader034.fdocuments.us/reader034/viewer/2022042408/5f22c0fa2d0ce76cb0441d3a/html5/thumbnails/72.jpg)
Thanks
THANK YOU FOR YOUR ATTENTION
Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov Finite simple quotients of groups satisfying property (T ).