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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 variety 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.

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.

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 the 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 variety 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.

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 variety 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.

I recently came up with 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 the 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 variety 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.

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 the 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 variety 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.