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8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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Juan González-Meneses
Universidad de Sevilla
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GDR Tresses et topologie de basse dimension
Île de Berder, November 15-17, 2006.
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8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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%RXQGLQJ WKH VL]H RI 866
In random examples of big canonical length, DOORIWKHP satisfy:
%UDLGJURXS %Q
*HEKDUGW This happens for Q=3,...,8 and
Remark:
$OOthese examples are SVHXGR$QRVRY and ULJLG
$OOthese examples are SVHXGR$QRVRY and ULJLG
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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PeriodicBraids = Roots of =, forsomem
ReducibleBraids = Preserve a family of disjoint, closed curves
1LHOVHQ7KXUVWRQ FODVVLILFDWLRQ
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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PeriodicBraids = Roots of =, forsomem
ReducibleBraids = Preserve a family of disjoint, closed curves
Pseudo-AnosovBraids = Preserve two transverse measured foliations…
…scaling the measure of by
and the measure of by
1LHOVHQ7KXUVWRQ FODVVLILFDWLRQ
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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PeriodicBraids = Roots of =, forsomem
ReducibleBraids = Preserve a family of disjoint, closed curves
Pseudo-AnosovBraids = Preserve two transverse measured foliations…
1LHOVHQ7KXUVWRQ FODVVLILFDWLRQ
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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PeriodicBraids = Roots of =, forsomem
ReducibleBraids = Preserve a family of disjoint, closed curves
Pseudo-AnosovBraids = Preserve two transverse measured foliations…
1LHOVHQ7KXUVWRQ FODVVLILFDWLRQ
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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3URSHUWLHV RI SHULRGLF EUDLGV
In general, #(USS( ; )) is exponential in Q.In general, #(USS( ; )) is exponential in Q.
%HVWYLQD (1999)
%LUPDQ*HEKDUGW*0 (2006)
%HVVLV'LJQH0LFKHO (2002)
The centralizer of ; is either %Q or the braid group of an annulus.The centralizer of ; is either %Q or the braid group of an annulus.
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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It can always be simplified by an automorphismK(i.e., by a conjugation).
3URSHUWLHV RI UHGXFLEOHEUDLGVA reducible braid D preserves a family of curves, called a UHGXFWLRQV\VWHP.
%LUPDQ/XERW]N\0F&DUWK\ (1983)
There is a FDQRQLFDOUHGXFWLRQV\VWHP
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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3URSHUWLHV RI UHGXFLEOHEUDLGV
Onecan thendecompose the disc D along
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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3URSHUWLHV RI UHGXFLEOHEUDLGV
Tubular braid:
Interior braids:
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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3URSHUWLHV RI UHGXFLEOHEUDLGV
*0 (2003):
*0:LHVW (2004):
The centralizer of D is a semi-direct product determined by:
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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3URSHUWLHV RI SVHXGR$QRVRY EUDLGV
A JHQHULFbraid is always pseudo-AnosovA JHQHULFbraid is always pseudo-Anosov (Not shown)
*0:LHVW (2004):
Its centralizer is isomorphic toIts centralizer is isomorphic to
%LUPDQ *HEKDUGW*0 (2006):
A small power of it is conjugate to a ULJLGEUDLG.A small power of it is conjugate to a ULJLGEUDLG. """
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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8VLQJ SRZHUV WR GHWHFW FRQMXJDF\
*0 (2003): The Pth root of a braid is unique up to conjugacy.
And if the braid is pseudo-Anosov, the root is unique.
&RUROODU\ ; and < are conjugate if and only if so are ; P and < P ; and < are conjugate if and only if so are ; P and < P
&RUROODU\
We can solve the CDP by using powers.
If ; and < are pseudo-Anosov, then the FRQMXJDWLQJHOHPHQWV
of ( ;< ) and of ( ; P , < P) coincide
If ; and < are pseudo-Anosov, then the FRQMXJDWLQJHOHPHQWV
of ( ;< ) and of ( ; P
, < P
) coincide
We can solve the CSP, LQWKHSVHXGR$QRVRYFDVH, by using powers.
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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5LJLG HOHPHQWV
In a Garside group:
In this case, cycling (& decycling) is just a cyclic permutation of the factors.
Hence ; USS( ; ), and its orbit under cycling has length either U or U.Hence ; USS( ; ), and its orbit under cycling has length either U or U.
(DVLHUFRPELQDWRULFV
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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5LJLG HOHPHQWV
%LUPDQ*HEKDUGW*0 (2006):
then DOOHOHPHQWV in USS( ; ) are rigid.
One can easily determine if an element is in USS( ; ).One can easily determine if an element is in USS( ; ).
Orbits under cycling are short.Orbits under cycling are short.
But how many orbits may appear in a USS?
2SHQSUREOHP
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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5LJLGLW\
If an element fails to be rigid, we can measure how far it is from being rigid.
The ULJLGLW\ of is:
N factors remain untouched
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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5LJLGLW\
([DPSOHV
All factors untouched
2/3’s of the factors untouched
No factor is untouched
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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&\FOLQJV DQG SRZHUV
Iterated cyclings and iterated powers of an element are closely related.
; SSS( ; ).
,WHUDWHGF\FOLQJV
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&\FOLQJV DQG SRZHUV
Iterated cyclings and iterated powers of an element are closely related
; SSS( ; ).
,WHUDWHGF\FOLQJV
,WHUDWHGSRZHUV
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&\FOLQJV DQG SRZHUV
Iterated cyclings and iterated powers of an element are closely related
; SSS( ; ).
,WHUDWHGF\FOLQJV
,WHUDWHGSRZHUV
%LUPDQ*HEKDUGW*0(2006)
The product of the first P factors in the left normal form of
is precisely
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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&\FOLQJV DQG SRZHUV([DPSOH
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&\FOLQJV DQG SRZHUV([DPSOH
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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&\FOLQJV DQG SRZHUV([DPSOH
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&\FOLQJV DQG SRZHUV([DPSOH
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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&\FOLQJV DQG SRZHUV([DPSOH
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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&RQVHTXHQFHV ; SSS( ; ).
&RUROODU\ Translation numbers in G are SRVLWLYH
Shown independentlyby /HH/HH (2006)
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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$EVROXWH IDFWRUV
; USS( ; ).
We studied the ILQDOIDFWRUV of:
)
)
)
)
) P
.
.
.
The chain stabilizes for
) ( ; ) Absolute final factor ) ( ; ) Absolute final factor
Looking at initial factors of suitable remainders
(ascending chain that stabilizes)
, ( ; ) Absolute initial factor , ( ; ) Absolute initial factor
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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$EVROXWH IDFWRUV
) ( ; ) , ( ; ) is always left-weighted. ) ( ; ) , ( ; ) is always left-weighted.
; USS( ; ).
Absolute factors are related to powers:
For every P z 1 such that ; P SSS( ; P), (For every P if ; belongsto the stable USS)
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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(OHPHQWV KDYLQJ DULJLG SRZHU
Suppose that ; USS( ; ) has a rigid power.
If an element has non-zero rigidity, its powers have the same inital factor.
Hence:
We need to know how many powers of ; may have 0 rigidity.
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(OHPHQWV KDYLQJ DULJLG SRZHU
([DPSOH
.
.
.
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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(OHPHQWV KDYLQJ DULJLG SRZHU
In this set there is a UHSHWLWLRQ
In the same way:
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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(OHPHQWV KDYLQJ DULJLG SRZHU
%XWZKLFKHOHPHQWVKDYHDULJLGSRZHU"
8/3/2019 Juan González-Meneses- Conjugacy problems in braid groups and other Garside groups Part 2
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SVHXGR$QRVRY EUDLGV
%LUPDQ*HEKDUGW*0 (2006)
0F&DUWK\ (1982), *0:LHVW (2004)
If ;is SVHXGR$QRVRY, every element in its FHQWUDOL]HU has a common powerwith ; , up to powers of '.
If 1is the orbit length of ; ,
Every pseudo-Anosov braid in its USS has a VPDOO power which is ULJLGEvery pseudo-Anosov braid in its USS has a VPDOO power which is ULJLG