Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F_q$ the unique field with $q$ elements.

What is known about the structure and properties of $\mathrm{SL}_2(F_q[t])$ as opposed to those of $\mathrm{SL}_2(\mathbb{Z})$? What do they have in common and what not?

Is this group virtually free as well?

Where can I read more about this group?

up vote 7 down vote accepted

One important thing that is known about the structure of $SL_2(\mathbb{F}_q[t])$ is Nagao's theorem: for any field $k$, in particular $k=\mathbb{F}_q$, there is an amalgam decomposition $$ SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t]), $$ where $B$ denotes upper triangular matrices. A good reference for statements about the structure of $SL_2(\mathbb{F}_q[t])$ would be Serre's book on trees. For instance, Nagao's theorem can be proved using the action of $SL_2(\mathbb{F}_q[t])$ on the associated Bruhat-Tits tree.

There are differences between the number-field situation and the function field situation, in particular where finiteness properties are concerned. While arithmetic groups in the number field situation do have good finiteness properties, $SL_2(\mathbb{F}_q[t])$ fails to be finitely generated - because of the infinite amalgam summand $B(\mathbb{F}_q[t])$. I think the group also fails to be virtually free for the same reason. (But the reason for this difference to the number field situation seems all concentrated at the prime $p$, somehow.)

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    There is quite a bit of related literature. The English translation of Serre's monograph is listed here: ams.org/mathscinet-getitem?mr=1954121, but there are also lots of papers such as ams.org/mathscinet-getitem?mr=0114866 and ams.org/mathscinet-getitem?mr=1177335 – Jim Humphreys Jan 23 '15 at 14:54
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    @JimHumphreys: right. The first MR-link is the paper in which Nagao's theorem was proven, it was later reinterpreted by Serre in the setting of group actions on trees. Since you mention the paper of Abels, there is a bulk of follow-up literature by Behr and more recently Bux, Wortmann,... on finiteness properties for $G(\mathbb{F}_q[t])$ with $G$ a Chevalley group (all by studying the actions of groups on Bruhat-Tits buildings). – Matthias Wendt Jan 23 '15 at 15:48
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    Another thing, related to the virtual freeness. It is always possible to get a finite index subgroup whose torsion elements all have $p$-power order, but it is not possible to get rid of the $p$-power torsion. So these groups are virtually $p'$-torsion free, but not virtually torsion-free. – Matthias Wendt Jan 23 '15 at 15:53
  • @MatthiasWendt In your answer, what does $k$ stand for? is it a certain field? – Pablo Jan 23 '15 at 17:02
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    @Matthias: you mean "any field, in particular for $k=\mathbb{F}_q$" (for the amalgam decomposition of $SL_2(k[t])$) – YCor Jan 23 '15 at 18:36

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