T HE PURE BRAID GROUPS AND THEIR RELATIVES Alex Suciu Northeastern University (joint work with He Wang) Séminaire d’algèbre et de géométrie Université de Caen January 5, 2016 A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 1 / 1
A RTIN ’ S BRAID GROUPS Let B n be the group of braids on n strings (under concatenation). B n is generated by σ 1 , . . . , σ n ´ 1 subject to the relations σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 and σ i σ j = σ j σ i for | i ´ j | ą 1. Let P n = ker ( B n ։ S n ) be the pure braid group on n strings. P n is generated by A ij = σ j ´ 1 ¨ ¨ ¨ σ i + 1 σ 2 i σ ´ 1 i + 1 ¨ ¨ ¨ σ ´ 1 j ´ 1 for 1 ď i ă j ď n . A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 2 / 1
B n = Mod 1 0 , n , the mapping class group of the 2-disk with n marked points. Thus, B n is a subgroup of Aut ( F n ) , and P n Ă IA n . In fact: B n = t β P Aut ( F n ) | β ( x i ) = wx τ ( i ) w ´ 1 , β ( x 1 ¨ ¨ ¨ x n ) = x 1 ¨ ¨ ¨ x n u . A classifying space for P n is the configuration space Conf n ( C ) = t ( z 1 , . . . , z n ) P C n | z i ‰ z j for i ‰ j u . Thus, B n = π 1 ( Conf n ( C ) / S n ) . Moreover, P n = F n ´ 1 ¸ α n ´ 1 P n ´ 1 = F n ´ 1 ¸ ¨ ¨ ¨ ¸ F 2 ¸ F 1 , where α n : P n Ă B n ã Ñ Aut ( F n ) . A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 3 / 1
W ELDED BRAID GROUPS The set of all permutation-conjugacy automorphisms of F n forms a subgroup of wB n Ă Aut ( F n ) , called the welded braid group. Let wP n = ker ( wB n ։ S n ) = IA n X wB n be the pure welded braid group wP n . McCool (1986) gave a finite presentation for wP n . It is generated by the automorphisms α ij (1 ď i ‰ j ď n ) sending x i ÞÑ x j x i x ´ 1 and j x k ÞÑ x k for k ‰ i , subject to the relations α ij α ik α jk = α jk α ik α ij for i , j , k distinct , [ α ij , α st ] = 1 for i , j , s , t distinct , [ α ik , α jk ] = 1 for i , j , k distinct . A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 4 / 1
The group wB n (respectively, wP n ) is the fundamental group of the space of untwisted flying rings (of unequal diameters), cf. Brendle and Hatcher (2013). Classical move Welded move The upper pure welded braid group (or, upper McCool group) is the subgroup wP + n Ă wP n generated by α ij for i ă j . We have wP + n – F n ´ 1 ¸ ¨ ¨ ¨ ¸ F 2 ¸ F 1 . P ROPOSITION (S.–W ANG ) For n ě 4 , the inclusion wP + Ñ wP n admits no splitting. n ã A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 5 / 1
V IRTUAL BRAID GROUPS The virtual braid group vB n is obtained from wB n by omitting certain commutation relations. Let vP n = ker ( vB n Ñ S n ) be the pure virtual braid group. Bardakov (2004) gave a presentation for vP n , with generators x ij for 1 ď i ‰ j ď n , i ´ 1 i + 1 j ´ 1 j j + 1 n i ´ 1 i + 1 j ´ 1 j j + 1 n 1 i 1 i ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ x ij x ji subject to the relations x ij x ik x jk = x jk x ik x ij , for i , j , k distinct , [ x ij , x st ] = 1 , for i , j , s , t distinct . A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 6 / 1
Let vP + n be the subgroup of vP n generated by x ij for i ă j . The inclusion vP + Ñ vP n is a split injection. n ã Bartholdi, Enriquez, Etingof, and Rains (2006) studied vP n and vP + n as groups arising from the Yang-Baxter equation. They constructed classifying spaces for these groups by taking quotients of permutahedra by suitable actions of the symmetric groups. A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 7 / 1
S UMMARY OF BRAID - LIKE GROUPS A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 8 / 1
C OHOMOLOGY RINGS AND B ETTI NUMBERS The cohomology algebras of the pure-braid like groups: H ˚ ( P n , C ) : Arnol’d (1969). H ˚ ( wP n , C ) : Jensen, McCammond, and Meier (2006). H ˚ ( wP + n ; C ) : F . Cohen, Pakhianathan, Vershinin, and Wu (2007) . H ˚ ( vP n ; C ) and H ˚ ( vP + n ; C ) : Bartholdi et al (2006), P . Lee (2013). The Betti numbers of the pure-braid like groups: wP + vP + P n wP n vP n n n ( n ´ 1 i ) n i b i s ( n , n ´ i ) s ( n , n ´ i ) L ( n , n ´ i ) S ( n , n ´ i ) Here s ( n , k ) are the Stirling numbers of the first kind, S ( n , k ) are the Stirling numbers of the second kind, and L ( n , k ) are the Lah numbers. A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 9 / 1
H ˚ ( P n ; C ) H ˚ ( wP n ; C ) H ˚ ( wP + n ; C ) H ˚ ( vP n ; C ) H ˚ ( vP + n ; C ) Generators u ij ( i ă j ) a ij ( i ‰ j ) e ij ( i ă j ) a ij ( i ‰ j ) e ij ( i ă j ) Relations (I1) (I2) (I3) (I5) (I2)(I3)(I4) (I5) (I6) No for Koszul Yes Yes Yes Yes n ě 4 u jk u ik = u ij ( u ik ´ u jk ) for i ă j ă k , (I1) a ij a ji = 0 for i ‰ j , (I2) a kj a ik = a ij ( a ik ´ a jk ) for i , j , k distinct , (I3) a ji a ik = ( a ij ´ a ik ) a jk for i , j , k distinct , (I4) (I5) e ij ( e ik ´ e jk ) = 0 for i ă j ă k , (I6) ( e ij ´ e ik ) e jk = 0 for i ă j ă k . Koszulness for P n : Arnol’d, Kohno. Koszulness for vP n and vP + n : Bartholdi et al (2006), Lee (2013). Koszulness for wP + n : D. Cohen and G. Pruidze (2008). Non-Koszulness for wP n : Conner and Goetz (2015). A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 10 / 1
A SSOCIATED GRADED L IE ALGEBRAS For a finitely generated group G , define the lower central series inductively by γ 1 G = G and γ k + 1 G = [ γ k G , G ] . The group commutator induces a graded Lie algebra structure on à gr ( G ) = k ě 1 ( γ k G / γ k + 1 G ) b Z C . gr ( vP + gr ( wP + n ) gr ( P n ) gr ( wP n ) n ) gr ( vP n ) Generators x ij , i ă j x ij , i ‰ j x ij , i ă j x ij , i ‰ j x ij , i ă j Relations L2, L4 L1, L2, L3 L1, L2, L3 L1, L2 L1, L2 Kohno, Bartholdi et al., Bartholdi et al., Jensen et al. F. Cohen et al. Falk–Randell Lee Lee [ x ij , x ik ] + [ x ij , x jk ] + [ x ik , x jk ] = 0 (L1) for distinct i , j , k , [ x ij , x kl ] = 0 for t i , j u X t k , l u = H , (L2) [ x ik , x jk ] = 0 for distinct i , j , k , (L3) [ x im , x ij + x ik + x jk ] = 0 for m = j , k and i , j , m distinct . (L4) A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 11 / 1
Let φ k ( G ) = dim gr k ( G ) be the LCS ranks of G . ř E.g.: φ k ( F n ) = 1 d | k µ ( k d ) n d . k By the Poincaré–Birkhoff–Witt theorem, 8 ź ( 1 ´ t k ) ´ φ k ( G ) = Hilb ( U ( gr ( G )) , t ) . k = 1 P ROPOSITION (P APADIMA –Y UZVINSKY 1999) Suppose gr ( G ) is quadratic and A = H ˚ ( G ; C ) is Koszul. Then Hilb ( U ( gr ( G )) , t ) ¨ Hilb ( A , ´ t ) = 1 . If G is a pure braid-like group, then gr ( G ) is quadratic. Furthermore, if G ‰ wP n ( n ě 4), then H ˚ ( G ; C ) is Koszul. Thus, ź 8 ÿ ( 1 ´ t k ) φ k ( G ) = b i ( G )( ´ t ) i . k = 1 i ě 0 A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 12 / 1
C HEN L IE ALGEBRAS The Chen Lie algebra of a f.g. group G is gr ( G / G 2 ) , the associated graded Lie algebra of its maximal metabelian quotient. Let θ k ( G ) = dim gr k ( G / G 2 ) be the Chen ranks of G . Easy to see: θ k ( G ) ď φ k ( G ) and θ k ( G ) = φ k ( G ) for k ď 3. Chen(1951): θ k ( F n ) = ( k ´ 1 )( n + k ´ 2 ) for k ě 2. k T HEOREM (D. C OHEN –S. 1993) The Chen ranks θ k = θ k ( P n ) are given by θ 1 = ( n 2 ) , θ 2 = ( n 3 ) , and θ k = ( k ´ 1 )( n + 1 4 ) for k ě 3 . C OROLLARY Let Π n = F n ´ 1 ˆ ¨ ¨ ¨ ˆ F 1 . Then P n fl Π n for n ě 4 , although both groups have the same Betti numbers and LCS ranks. A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 13 / 1
T HEOREM (D. C OHEN –S CHENCK 2015) 2 ) + ( k 2 ´ 1 )( n θ k ( wP n ) = ( k ´ 1 )( n 3 ) , for k " 0 . T HEOREM (S.–W ANG ) The Chen ranks θ k = θ k ( wP + n ) are given by θ 1 = ( n 2 ) , θ 2 = ( n 3 ) , and ÿ k � n + i ´ 2 � � n + 1 � θ k = + , for k ě 3 . i + 1 4 i = 3 C OROLLARY wP + n fl P n and wP + n fl Π n for n ě 4 , although all three groups have the same Betti numbers and LCS ranks. This answers a question of F . Cohen et al. (2007). For n = 4, an incomplete argument was given by Bardakov and Mikhailov (2008), using single-variable Alexander polynomials. A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 14 / 1
R ESONANCE VARIETIES Let G be a finitely presented group, and set A = H ˚ ( G , C ) . The (first) resonance variety of G is given by R 1 ( G ) = t a P A 1 | D b P A 1 z C ¨ a such that a ¨ b = 0 P A 2 u . For instance, R 1 ( F n ) = C n for n ě 2, and R 1 ( Z n ) = t 0 u . T HEOREM (D. C OHEN –S. 1999) R 1 ( P n ) is a union of ( n 3 ) + ( n 4 ) linear subspaces of dimension 2 . T HEOREM (D. C OHEN 2009) R 1 ( wP n ) is a union of ( n 2 ) linear subspaces of dimension 2 and ( n 3 ) linear subspaces of dimension 3 . A LEX S UCIU (N ORTHEASTERN ) P URE BRAID GROUPS AND THEIR RELATIVES C AEN , J ANUARY 2016 15 / 1
Recommend
More recommend
