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1
2
SIMPLE QUESTION
SL2
- −1,−1
Z
Property (FA) of the unit group of 2 -by- 2 matrices over an order - - PowerPoint PPT Presentation
Property (FA) of the unit group of 2 -by- 2 matrices over an order - - PowerPoint PPT Presentation
Property (FA) of the unit group of 2 -by- 2 matrices over an order in a quaternion algebra Ann Kiefer joint work with Bchle, Janssens, Jespers and Temmerman 10 June 2019 1 SIMPLE QUESTION SL 2 1 , 1 Z 2 SIMPLE QUESTION
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SLIDE 3
2
SIMPLE QUESTION
SL2
- −1,−1
Z
- acts on
Hautes Fagnes / High Fens
SLIDE 4
2
SIMPLE QUESTION
SL2
- −1,−1
Z
- acts on
Hautes Fagnes / High Fens
Does the action have a fixed point?
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3
SERRE’S PROPERTY (FA)
POINTS FIXES SUR LES ARBRES
Definition
A group Γ is said to have property (FA) if every action on a tree has a fixed point.
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3
SERRE’S PROPERTY (FA)
POINTS FIXES SUR LES ARBRES
Definition
A group Γ is said to have property (FA) if every action on a tree has a fixed point.
Theorem (Serre)
A finitely generated group Γ has property (FA) if and only if it satisfies the following properties
◮ Γ has finite abelianization, ◮ Γ has no non-trivial decomposition as amalgamated product.
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THE HEREDITARY VERSION OF (FA)
K subgroup of finite index in Γ K has (FA) ⇒ Γ has (FA) Problem: Γ has (FA) ✟
✟
⇒ K has (FA)
Definition (Property (HFA))
A group Γ is said to have property (HFA) if every finite index subgroup has property (FA).
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MOTIVATION
When does U(ZG) has (FA)? When does U(ZG) has (HFA)?
Cases that have to be handled
◮ SL2(O2), with O2 maximal order in
- −1,−1
Q
- ◮ SL2(O3), with O3 maximal order in
- −1,−3
Q
- ◮ SL2(O5), with O5 maximal order in
- −2,−5
Q
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BACK TO THE BEGINNING
◮ SL2(O2), with O2 maximal order in
- −1,−1
Q
- Easier: SL2
- −1,−1
Z
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BACK TO THE BEGINNING
◮ SL2(O2), with O2 maximal order in
- −1,−1
Q
- Easier: SL2
- −1,−1
Z
- SL2
- −1,−1
Z
- acts on
Does the action have a fixed point?
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STEP BY STEP
known: SL2(Z) = C4 ∗C2 C6 known: SL2(Z[i]) = G1 ∗SL2(Z) G2
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STEP BY STEP
known: SL2(Z) = C4 ∗C2 C6 known: SL2(Z[i]) = G1 ∗SL2(Z) G2 Generalization: Vahlen Group SL+(Γn(Z)) n=1 SL2(Z) n=2 SL2(Z[i]) n=4 subgroup of finite index in SL2
- −1,−1
Z
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STEP BY STEP
known: SL2(Z) = C4 ∗C2 C6 known: SL2(Z[i]) = G1 ∗SL2(Z) G2 Generalization: Vahlen Group SL+(Γn(Z)) n=1 SL2(Z) n=2 SL2(Z[i]) n=4 subgroup of finite index in SL2
- −1,−1
Z
- Theorem (Bächle-Janssens-Jespers-K.-Temmerman)
SL+(Γ3(Z)) = H1 ∗SL2(Z[i]) H2 SL+(Γ4(Z)) = K1 ∗SL+(Γ3(Z)) K2 → SL2
- −1,−1
Z
- does not have (HFA)
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WHAT ABOUT THE OTHER QUATERNION ALGEBRAS?
E2 group generated by elementary matrices E2 has finite index in SL2(O) O maximal order in
- −1,−1
Q
- −1,−3
Q
- −2,−5
Q
- E2(O) finite abelianization
E2(O) amalgamated product E2(O) (FA) E2(O) (HFA)
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WHAT ABOUT THE OTHER QUATERNION ALGEBRAS?
E2 group generated by elementary matrices E2 has finite index in SL2(O) O maximal order in
- −1,−1
Q
- −1,−3
Q
- −2,−5
Q
- E2(O) finite abelianization
E2(O) amalgamated product E2(O) (FA) E2(O) (HFA) ×
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WHAT ABOUT THE OTHER QUATERNION ALGEBRAS?
E2 group generated by elementary matrices E2 has finite index in SL2(O) O maximal order in
- −1,−1
Q
- −1,−3
Q
- −2,−5
Q
- E2(O) finite abelianization
- ×
E2(O) amalgamated product E2(O) (FA) E2(O) (HFA) ×
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WHAT ABOUT THE OTHER QUATERNION ALGEBRAS?
E2 group generated by elementary matrices E2 has finite index in SL2(O) O maximal order in
- −1,−1
Q
- −1,−3
Q
- −2,−5
Q
- E2(O) finite abelianization
- ×
E2(O) amalgamated product × × ? E2(O) (FA) E2(O) (HFA) ×
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WHAT ABOUT THE OTHER QUATERNION ALGEBRAS?
E2 group generated by elementary matrices E2 has finite index in SL2(O) O maximal order in
- −1,−1
Q
- −1,−3
Q
- −2,−5
Q
- E2(O) finite abelianization
- ×
E2(O) amalgamated product × × ? E2(O) (FA)
- ×
E2(O) (HFA) ×
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WHAT ABOUT THE OTHER QUATERNION ALGEBRAS?
E2 group generated by elementary matrices E2 has finite index in SL2(O) O maximal order in
- −1,−1
Q
- −1,−3
Q
- −2,−5
Q
- E2(O) finite abelianization
- ×
E2(O) amalgamated product × × ? E2(O) (FA)
- ×
E2(O) (HFA) × × ×
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WHAT ABOUT THE OTHER QUATERNION ALGEBRAS?
E2 group generated by elementary matrices E2 has finite index in SL2(O) O maximal order in
- −1,−1
Q
- −1,−3
Q
- −2,−5
Q
- E2(O) finite abelianization
- ×
E2(O) amalgamated product × × ? E2(O) (FA)
- ×
E2(O) (HFA) × × × SL2(O) (FA)
- ?
SL2(O) (HFA) × × ×
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TRAILER
When does U(ZG) has (FA)? When does U(ZG) has (HFA)?
Tomorrow 11:00 Doryan’s talk
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