# Demazure’s principle for tower of reductive groups and analogy with Teichmuller tower

Grothendieck in his Esquisse states (page 6/7 of the English translation) that the “principle” that the Teichmuller tower, i.e., the system of profinite fundamental groupoids $\hat{T}_{g,\nu}$ of moduli spaces of $\nu$-pointed genus $g$ curves $M_{g,\nu}$ under the operation of glueing can be reconstituted from generators in level ($= 3g - 3 + \nu$) 1 and relations in level 2 is strikingly analogous to Demazure’s principle for reconstitution of all reductive groups from those at level (= semi-simple rank) 1 and 2. He also thinks that the analogy is not just formal.

My question is: where can one find an exposition of Demazure’s principle? Also, have there been any further developments on Grothendieck’s analogy?

• Presumably "reductive" really means "split reductive" (or maybe Grothendieck has in mind to be working over $\mathbf{C}$, rendering the two notions equivalent), and Demazure's principle (though I've never heard it referred to as such before) is probably SGA3, Exp. XXIII, 2.3 or more likely 2.4 (also see 2.6, 3.1.3, 3.2.8, 3.3.7, 3.4.10, 3.5 in Exp. XXIII). I've never seen elsewhere a comparably clean exposition along the lines given there, even if just limited to working over $\mathbf{C}$. – nfdc23 Mar 24 '18 at 23:29