Morphisms between free groups.

[Michel VAN DEN BERGH]

Dear all,

I need to do something which I thought would be quite simple. I have free groups

F, G with generators x1,x2,x3,x4 and x,y respectively and I need to compute the image of an element under the homomorphism F-->G:x1->x, x2->y, x3->x, x4->y.

I could not find anything in the manual about morphisms between free groups. I tried subs, but that does not work since the parents of the elements of F and G are (of course) different.

The workaround I used is to work in the free group H with generators x1,x2,x3,x4,x,y and to use subs. However this is also inconvenient since I need the action of the braid group with 4 strands on F. It does not act on H because of the limitation "#strands=#generators".

So this means I have to use only part of the braid group with 6 strands.

In the end it all works but it is terribly hacky for some which seems to be a very clean thing to do.

I would be grateful for any suggestions.

Michel

 

[David Joyner]

Michel:

One idea is to check out
https://doc.sagemath.org/html/en/reference/groups/sage/groups/braid.html

Another idea is to download braid-1.1.tar.gz, untar in the pkg
subdirectory and type LoadPackage("braid");
FYI, the pkg subdir is inside sage-9*/local/share/gap
You can find a pdf manual for the braid package at
https://arxiv.org/abs/math/0304376
Currently, the link to braid-1.1.tar.gz at
https://www.gap-system.org/Packages/undep.html
is bad but I have a copy. Just email me privately.

- David

> Michel
>
>
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[Dima Pasechnik]

On Fri, 4 Feb 2022, 11:02 David Joyner, <wdjo...@gmail.com> wrote:

On Fri, Feb 4, 2022 at 4:40 AM 'Michel VAN DEN BERGH' via sage-support
<sage-s...@googlegroups.com> wrote:
>
> Dear all,
>
> I need to do something which I thought would be quite simple. I have free groups
> F, G with generators x1,x2,x3,x4 and x,y respectively and I need to compute the image of an element under the homomorphism F-->G:x1->x, x2->y, x3->x, x4->y.
>
> I could not find anything in the manual about morphisms between free groups. I tried subs, but that does not work since the parents of the elements of F and G are (of course) different.
>
> The workaround I used is to work in the free group H with generators x1,x2,x3,x4,x,y and to use subs. However this is also inconvenient since I need the action of the braid group with 4 strands on F. It does not act on H because of the limitation "#strands=#generators".
>
> So this means I have to use only part of the braid group with 6 strands.
>
> In the end it all works but it is terribly hacky for some which seems to be a very clean thing to do.
>
> I would be grateful for any suggestions.

Michel:

One idea is to check out
https://doc.sagemath.org/html/en/reference/groups/sage/groups/braid.html

Another idea is to download braid-1.1.tar.gz, untar in the pkg
subdirectory and type LoadPackage("braid");

most, if not all, functionality of braid is nowadays in

https://gap-packages.github.io/MapClass/

(which is in our gap_packages)

FYI, the pkg subdir is inside sage-9*/local/share/gap
You can find a pdf manual for the braid package at
https://arxiv.org/abs/math/0304376
Currently, the link to braid-1.1.tar.gz at
https://www.gap-system.org/Packages/undep.html
is bad but I have a copy. Just email me privately.

- David

> Michel
>
>
> --
> You received this message because you are subscribed to the Google Groups "sage-support" group.
> To unsubscribe from this group and stop receiving emails from it, send an email to sage-support...@googlegroups.com.
> To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/f7d460d0-390b-455d-ae90-72b105e8e926n%40googlegroups.com.

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[Michel VAN DEN BERGH]

Now it seems "subs" does work. I must have done something wrong. Sorry for the noise.