Return to Answer

See the paper link.springer.com/article/10.1007%2Fs00222-013-0477-9; the action of the pure braid group on the $abelianisation$ of the derived group is described there (see Section(3.4). With some qualification, it is indeed "the" Gassner representation. The trouble is that $F'/F''$ is not a free module over the group algebra ${\mathbb Z}[F/F']$, and hence one embeds the quotient $F'/F''$ into a slightly larger module which is indeed free and which is is exactly the Gassner representation; all this is worked out in the paper I have referred to.

I just now saw the second part of your question: the action of $P$ is indeed highly nontrivial on $F'/F''$. The image is (modulo $d$-th roots of unity, with $n\geq 2d-1$ if you are dealing with the pure braid group on $n$ strands) an arithmetic group in the unitary group associated to the Gassner representation

I could not copy the link so here is the reference:

http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=JOUR&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Venkataramana%2CT%2A&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=3219513

See the paper link.springer.com/article/10.1007%2Fs00222-013-0477-9; the action of the pure braid group on the $abelianisation$ of the derived group is described there (see Section(3.4). With some qualification, it is indeed "the" Gassner representation. The trouble is that $F'/F''$ is not a free module over the group algebra ${\mathbb Z}[F/F']$, and hence one embeds the quotient $F'/F''$ into a slightly larger module which is indeed free and which is is exactly the Gassner representation; all this is worked out in the paper I have referred to.

I just now saw the second part of your question: the action of $P$ is indeed highly nontrivial on $F'/F''$. The image is (modulo $d$-th roots of unity, with $n\geq 2d-1$ if you are dealing with the pure braid group on $n$ strands) an arithmetic group in the unitary group associated to the Gassner representation

I could not copy the link so here is the reference: http://www.ams.org/mathscinet-getitem?mr=3219513

See the paper link.springer.com/article/10.1007%2Fs00222-013-0477-9; the action of the pure braid group on the $abelianisation$ of the derived group is described there (see Section(3.4). With some qualification, it is indeed "the" Gassner representation. The trouble is that $F'/F''$ is not a free module over the group algebra ${\mathbb Z}[F/F']$, and hence one embeds the quotient $F'/F''$ into a slightly larger module which is indeed free and which is is exactly the Gassner representation; all this is worked out in the paper I have referred to.

I could not copy the link so here is the reference:

http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=JOUR&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Venkataramana%2CT%2A&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=3219513

See the paper link.springer.com/article/10.1007%2Fs00222-013-0477-9; the action of the pure braid group on the $abelianisation$ of the derived group is described there (see Section(3.4). With some qualification, it is indeed "the" Gassner representation. The trouble is that $F'/F''$ is not a free module over the group algebra ${\mathbb Z}[F/F']$, and hence one embeds the quotient $F'/F''$ into a slightly larger module which is indeed free and which is is exactly the Gassner representation; all this is worked out in the paper I have referred to.

I just now saw the second part of your question: the action of $P$ is indeed highly nontrivial on $F'/F''$. The image is (modulo $d$-th roots of unity, with $n\geq 2d-1$ if you are dealing with the pure braid group on $n$ strands) an arithmetic group in the unitary group associated to the Gassner representation

I could not copy the link so here is the reference:

http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=JOUR&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Venkataramana%2CT%2A&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=3219513

See the paper link.springer.com/article/10.1007%2Fs00222-013-0477-9; the action of the pure braid group on the $abelianisation$ of the derived group is described there (see Section(3.4). With some qualification, it is indeed "the" Gassner representation. I could not copy the link so here is the reference:

http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=JOUR&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Venkataramana%2CT%2A&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=3219513

See the paper link.springer.com/article/10.1007%2Fs00222-013-0477-9; the action of the pure braid group on the $abelianisation$ of the derived group is described there (see Section(3.4). With some qualification, it is indeed "the" Gassner representation. The trouble is that $F'/F''$ is not a free module over the group algebra ${\mathbb Z}[F/F']$, and hence one embeds the quotient $F'/F''$ into a slightly larger module which is indeed free and which is is exactly the Gassner representation; all this is worked out in the paper I have referred to.

I could not copy the link so here is the reference:

http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=JOUR&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Venkataramana%2CT%2A&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=3219513