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Quomodocumque on F_un (Update 10-21)

Posted By Thas On October 14, 2008 @ 2:04 pm In news | No Comments

Quomodocumque has an interesting post on F_un, buildings, the braid group and GL_n(\mathbb{F}_1[t,t^{-1}]) [1], starting from the Kapranov-Smirnov observation that GL_n(\mathbb{F}_1[t]) should be considered as the braid group on n-strings (see also this post [2]).

So then, what is GL_n(\mathbb{F}_1[t,t^{-1}])? For n=2 it turns out that one can use Serre’s-tree results in the description of GL_2(\mathbb{F}_q[t,t^{-1}]) to postulate that

GL_2(\mathbb{F}_1[t,t^{-1}]) = \Z \ast_{2 \Z} \Z, the amalgamated free product of two copies over \Z over the common subgroup 2\Z.

For higher n the work of Lisa Carbone [3] suggest the following answer :

GL_n(\mathbb{F}_1[t,t^{-1}]) is something like the mapping class group of a disc with n boundary components instead of n punctures.”

UPDATE tuesday october 20th

In a follow-up post, F_un and the braid group – a note of skepticism [4], arguments are collected against the claim that the d-string braid group B_d should be GL_d(\mathbb{F}_1[t]). One of the arguments being that GL_d(\mathbb{F}_1) is a subgroup of GL_d(\mathbb{F}_1[t]) whereas S_d cannot be embedded into B_d.


Article printed from F_un mathematics: http://cage.ugent.be/~kthas/Fun

URL to article: http://cage.ugent.be/~kthas/Fun/index.php/quomodocumque-on-f_un.html

URLs in this post:

[1] Image: http://quomodocumque.wordpress.com/2008/10/13/f_1-buildings-the-braid-group-gl_nf_1t1t/

[2] this post: http://cage.ugent.be/~kthas/Fun/index.php/f_un-and-braid-groups.html

[3] Lisa Carbone: http://www.math.rutgers.edu/~carbonel/

[4] F_un and the braid group – a note of skepticism: http://quomodocumque.wordpress.com/2008/10/20/f_1-and-the-braid-group-a-note-of-skepticism/

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