# Decomposition of linear groups into free products

I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of Serre generalizes this theorem to the situation that you can take $GL_2$ of the coordinate ring of a smooth projective curve minus a point and it will be equal to the fundamental group of a graph group.

I wonder whether there are higher dimensional versions of this theorem or not, in the sense that you can both consider $GL_n$ instead of $GL_2$ and higher dimensional varieties instead of curves. I would really appreciate it if you could point me to some references about these kinds of theorems.

• Note that $\text{SL}_3(\mathbb{Z})$ and $\text{SL}_3(\mathbb{F}_p[t])$ have Serre's property FA: every action on a tree has a fixed point. – Uri Bader Aug 20 '18 at 9:19
• For GL_n, or more generally a Chevalley group, there is a generalisation of Nagao's theorem by C.Soule, "Chevalley groups over polynomial rings". – naf Aug 20 '18 at 9:36
• @ulrich Thanks that was helpful but still Soule works over polynomial rings with one variable. I'm not sure but one might able to generalized his work for curves but I imagine things doesn't work well for higher dimensional varieties. In my opinion local fields seems to be an essential part of the Bruhat-Tits building which you have access to them only when you work with curves over finite fields. I suspect one might need Parshin's Bruhat-Tits buildings for higher dimensional local fields for general varieties, which looks like they are not well studied. – user127776 Aug 20 '18 at 16:32
• For higher dimensional varieties, one ought to be able to take valuations associated to ideal points and get actions on Bruhat-Tits buildings, and hence presentations as a complex of groups (generalizing the notion of a graph of groups). However, I don't know a reference immediately. – Ian Agol Aug 20 '18 at 17:57