Three theorems about generating mapping class...
Transcript of Three theorems about generating mapping class...
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reducible, nonperiodic reducible, periodic
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!?
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!?
Tara Brendle and Benson Farb
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!?
!!??
Tara Brendle and Benson Farb
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There are numbers bigger than 2.
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Order of elements
Number of elements
Genus
Brendle-Farb 2 6 g ≥ 3
Kassabov 2 4 g ≥ 7
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Order of elements
Number of elements
Genus
Brendle-Farb 2 6 g ≥ 3
Kassabov 2 4 g ≥ 7
Monden 3 3 g ≥ 3
4 4 g ≥ 3
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Order of elements
Number of elements
Genus
Brendle-Farb 2 6 g ≥ 3
Kassabov 2 4 g ≥ 7
Monden 3 3 g ≥ 3
4 4 g ≥ 3
Yoshihara 6 3 g ≥ 10
6 4 g ≥ 5
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Obstacle:When do higher-order elements even exist in
Mod(Sg)?
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Orders of torsion elements in Mod(S3):
1, 2, 3, 4, 6, 7, 8, 9, 12, 14
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Number theoretic conditions for the existence of torsion elements
in Mod(Sg)
(Ashikaga and Ishizaka)
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Strategy:
1) Find some order 2 elements.2) Write a Dehn twist as a product in these.3) Show that the order 2 elements generate a subgroup that puts all the Humphries curves in the same orbit.
(Luo, Brendle-Farb)
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1) Find some order 2 elements.2) Write a Dehn twist as a product in these.3) Show that the order 2 elements generate a subgroup that puts all the Humphries curves in the same orbit.
Strategy:(Luo, Brendle-Farb)
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(Luo)
Td = (TxTa-1)(TyTb
-1)(TzTc-1)
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(Luo)
TxTa-1 = Tx(f Tx
-1f-1) = (Txf Tx-1)f-1
x a
f
Td = (TxTa-1)(TyTb
-1)(TzTc-1)
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(Brendle-Farb)
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(Brendle-Farb)
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Strategy:
1) Find some order k elements.2) Write a Dehn twist as a product in these.3) Show that the order k elements generate a subgroup that puts all the Humphries curves in the same orbit.
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genus 4genus 5
k = 5
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TxTa-1 = Tx(f Tx
-1f-1) = (Txf Tx-1)f-1
Td = (TxTa-1)(TyTb
-1)(TzTc-1)
f xa
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Other groups?
Sharpening these results?
Other periodic elements?
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reducible elements
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reducible elements
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periodic elements
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periodic elements
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Obstacle:How can you even get a handle
on all the periodic elements?
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Obstacle
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