MAPPING CLASS GROUPS

These are problems about mapping class groups, with a focus on braid groups. See also problems (O10), (E4).

*(B1) Are braid groups linear? Background

(B2) Is the Gassner representation of the pure braid group P_n faithful for every n ? Background

(B3) Is the Burau representation of the braid group B_n faithful for n = 4 ? Background

(B4) (J.Birman) Give necessary and sufficient conditions for a square matrix over Laurent polynomial ring to be the Burau matrix of some braid.

(B5) (V.Lin) Let n > 4. (a) Does the braid group B_n have a non-trivial non-injective endomorphism?

(b) Is it true that every non-trivial endomorphism of the commutator subgroup [B_n, B_n] is actually an automorphism ? Background

* (B6) (V.Lin) Let n > 4.  (a) Does the braid group B_n have a proper torsion-free non-abelian factor group?

(b) Does the commutator subgroup [B_n, B_n] have a proper torsion-free factor group? Background

(B7) (V.Lin) Let n > 4. (a) Is it true that every automorphism of the commutator subgroup [B_n, B_n] can be extended to an automorphism of the whole group B_n ?

(b) Is it true that every non-trivial endomorphism of [B_n, B_n] can be extended to an endomorphism of B_n ? Background

(B8) (P. Dehornoy) We call a braid word  w   \sigma-positive (resp.  \sigma-negative) if the generator  \sigma_i  with minimal index occurs only with  positive (resp.  negative) exponents  in  w.   Is it true that, for every  n,  there exists a constant  c(n)  such that every  n-strand braid word of length  N  is equivalent to a  \sigma-positive or a  \sigma-negative braid word of length  c(n) N  at most ?

(B9) (P. Dehornoy)  Say that a braid is special if it can be obtained from the trivial braid by using iteratively the self-distributive exponentiation  a \wedge b = a  S(b)  sigma_1 S(a)^{-1},   where  S  is the shift endomorphism that maps  \sigma_i to  \sigma_{i+1}  for every  i.  How many special braids are there in  B_n?

*(B10) (G.Makanin) Is it true that in any group B_n,   g^k = h^k  for some  k \ne 0  implies that g is conjugate to h ? Background

(B11) (V.Shpilrain) Let K_n be the kernel of the Burau representation of the braid group B_n,  n>4. Is the factor group  B_n/K_n torsion free? Background

(B12) Do braid groups B_n,  n>4, have non-elementary hyperbolic factor groups? Background