Quomodocumque on F_un (Update 10-21)

Posted by on Oct 14, 2008 in newsNo comments

Quomodocumque has an interesting post on F_un, buildings, the braid group and GL_n(\mathbb{F}_1[t,t^{-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).

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 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, 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.

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