A LGEBRAIC INVARIANTS OF PURE BRAID - LIKE GROUPS Alex Suciu Northeastern University (joint work with He Wang) Workshop on Braids in Algebra, Geometry, and Topology International Centre for Mathematical Sciences, Edinburgh, UK May 23, 2017 A LEX S UCIU (N ORTHEASTERN ) P URE BRAID - LIKE GROUPS E DINBURGH , M AY 2017 1 / 22
P URE - BRAID LIKE GROUPS B RAID GROUPS 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 (1 ď i ă j ď n ). A LEX S UCIU (N ORTHEASTERN ) P URE BRAID - LIKE GROUPS E DINBURGH , M AY 2017 2 / 22
P URE - BRAID LIKE GROUPS B RAID GROUPS 0 , n , the mapping class group of D 2 with n marked points. B n = Mod 1 Thus, B n is a subgroup of Aut ( F 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 . P n is a subgroup of IA n = t ϕ P Aut ( F n ) | ϕ ˚ = id on H 1 ( F 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 - LIKE GROUPS E DINBURGH , M AY 2017 3 / 22
P URE - BRAID LIKE GROUPS W ELDED BRAID GROUPS 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 - LIKE GROUPS E DINBURGH , M AY 2017 4 / 22
P URE - BRAID LIKE GROUPS W ELDED BRAID GROUPS 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). 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 . L EMMA (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 - LIKE GROUPS E DINBURGH , M AY 2017 5 / 22
P URE - BRAID LIKE GROUPS V IRTUAL BRAID GROUPS 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 (1 ď i ‰ j ď n ), 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 . 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 by taking quotients of permutahedra by suitable actions of the symmetric groups. A LEX S UCIU (N ORTHEASTERN ) P URE BRAID - LIKE GROUPS E DINBURGH , M AY 2017 6 / 22
P URE - BRAID LIKE GROUPS V IRTUAL BRAID GROUPS S UMMARY OF BRAID - LIKE GROUPS A LEX S UCIU (N ORTHEASTERN ) P URE BRAID - LIKE GROUPS E DINBURGH , M AY 2017 7 / 22
C OHOMOLOGY RINGS AND L IE ALGEBRAS C OHOMOLOGY RINGS C OHOMOLOGY RINGS AND B ETTI NUMBERS Arnol’d (1969): H ˚ ( P n ) = Ź i ă j ( e ij ) / x e jk e ik ´ e ij ( e ik ´ e jk ) y . Jensen, McCammond, and Meier (2006): H ˚ ( wP n ) = Ź i ‰ j ( e ij ) / x e ij e ji , e jk e ik ´ e ij ( e ik ´ e jk ) y . F . Cohen, Pakhianathan, Vershinin, and Wu (2007): n ) = Ź H ˚ ( wP + i ă j ( e ij ) / x e ij ( e ik ´ e jk ) y . Bartholdi et al (2006), P . Lee (2013): H ˚ ( vP n ) = Ź i ‰ j ( e ij ) / x e ij e ji , e ij ( e ik ´ e jk ) , e ji e ik = ( e ij ´ e ik ) e jk ) y , n ) = Ź H ˚ ( vP + i ă j ( e ij ) / x e ij ( e ik ´ e jk ) , ( e ij ´ e ik ) e jk y . All these Q -algebras A are quadratic. In fact, they are all Koszul algebras (Tor A i ( Q , Q ) j = 0 for i ‰ j ), except for H ˚ ( wP n ) , n ě 4. P n : Kohno (1987). wP n : Conner and Goetz (2015). wP + n : D. Cohen and G. Pruidze (2008). vP n and vP + n : Bartholdi et al (2006), Lee (2013). A LEX S UCIU (N ORTHEASTERN ) P URE BRAID - LIKE GROUPS E DINBURGH , M AY 2017 8 / 22
C OHOMOLOGY RINGS AND L IE ALGEBRAS C OHOMOLOGY RINGS The Betti numbers of the pure-braid like groups are given by 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 - LIKE GROUPS E DINBURGH , M AY 2017 9 / 22
C OHOMOLOGY RINGS AND L IE ALGEBRAS A SSOCIATED GRADED AND HOLONOMY L IE ALGEBRAS A SSOCIATED GRADED L IE ALGEBRAS The lower central series of a group G is defined 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 gr ( G ) = À k ě 1 ( γ k G / γ k + 1 G ) b Z Q Assume G is finitely generated. Then gr ( G ) is also finitely generated: in degree 1, by gr 1 ( G ) = H 1 ( G , Q ) . Let A ˚ = H ˚ ( G , Q ) , let µ A : A 1 ^ A 1 Ñ A 2 be the cup-product map, and µ _ A : A 2 Ñ A 1 ^ A 1 its dual, where A i = ( A i ) _ . Define the holonomy Lie algebra h ( G ) : = h ( A ) as the quotient Lie ( A 1 ) by the ideal generated by im ( µ _ A ) Ă A 1 ^ A 1 = Lie 2 ( A 1 ) . There is a canonical surjection h ( G ) ։ gr ( G ) which is an isomorphism precisely when gr ( G ) is quadratic. A LEX S UCIU (N ORTHEASTERN ) P URE BRAID - LIKE GROUPS E DINBURGH , M AY 2017 10 / 22
C OHOMOLOGY RINGS AND L IE ALGEBRAS A SSOCIATED GRADED AND HOLONOMY L IE ALGEBRAS 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 ; Q ) is Koszul. Then Hilb ( U ( gr ( G )) , t ) ¨ Hilb ( A , ´ t ) = 1 . Let G be a pure braid-like group. Then gr ( G ) is quadratic. Furthermore, if G ‰ wP n ( n ě 4), then H ˚ ( G ; Q ) 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 - LIKE GROUPS E DINBURGH , M AY 2017 11 / 22
C OHOMOLOGY RINGS AND L IE ALGEBRAS C HEN L IE ALGEBRAS 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. K.-T. 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 - LIKE GROUPS E DINBURGH , M AY 2017 12 / 22
C OHOMOLOGY RINGS AND L IE ALGEBRAS C HEN L IE ALGEBRAS 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). A LEX S UCIU (N ORTHEASTERN ) P URE BRAID - LIKE GROUPS E DINBURGH , M AY 2017 13 / 22
C OHOMOLOGY RINGS AND L IE ALGEBRAS R ESONANCE VARIETIES R ESONANCE VARIETIES Let A be a graded C -algebra with A 0 = C and dim A 1 ă 8 . The (first) resonance variety of A is defined as R 1 ( A ) = t a P A 1 | D b P A 1 z C ¨ a such that a ¨ b = 0 P A 2 u . For a finitely generated group G , define R 1 ( G ) : = R 1 ( H ˚ ( G ; C )) . For instance, R 1 ( F n ) = C n for n ě 2, and R 1 ( Z n ) = t 0 u . P ROPOSITION (D. C OHEN –S. 1999) R 1 ( P n ) is a union of ( n 3 ) + ( n 4 ) linear subspaces of dimension 2 . P ROPOSITION (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 - LIKE GROUPS E DINBURGH , M AY 2017 14 / 22
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