Applications of the fixed point theorem for group actions on buildings to algebraic groups over polynomial rings (Q6568808)

The authors apply the fixed point theorem for CAT(0) spaces, which states that any group of isometries of a complete CAT(0) space which stabilizes a non-empty bounded subset must also fix a point, to Bruhat-Tits buildings in order to show that the finite conjugacy property holds for groups of the form \(G(k[t])\) where \(G\) is a reductive algebraic group over a finite extension \(k\) of the \(p\)-adic field \(\mathbb{Q}_p\). In detail, their main result is the following:\N\NTheorem A: Let \(G\) be a reductive algebraic group defined over a finite extension \(k\) of the \(p\)-adic field \(\mathbb{Q}_p\). Then the group \(G(k[t])\) has finitely many conjugacy classes of finite subgroups.\N\NThe authors also highlight the following result for general fields of characteristic \(0\) which is used in the proof of Theorem A.\N\NTheorem B: Let \(G\) be a reductive algebraic group over a field \(k\) of characteristic \(0\). Then every finite subgroup of \(G(k[t])\) is conjugate to a subgroup contained in \(G(k)\).\N\NThe majority of the paper lies in Section 2, which contains the heavy lifting provided by the theory of buildings. Here, the authors begin by setting notation and reviewing some previous results and then prove various decomposition results for the group \(G(k[t])\) by considering the building associated to the group \(G\times_k k((1/t))\). They also show that \(G(k[t])\) acts on the building by \emph{type-preserving automorphisms}, which are those that respect a particular coloring of the vertices of the building. For a Galois extension \(\ell/k\), they consider the natural action of \(\mathrm{Gal}(\ell/k)\) on the building associated with the group \(G\times_k \ell((1/t))\) and prove decomposition results about the pointwise stabilizer subgroups of invariant subsets of the building. The much shorter Section 3 contains some cohomological facts about semi-direct products of groups, in particular focusing on some cases when a finite subgroup of \(N\rtimes H\) must be conjugate to a subgroup of \(H\). The proofs of the two main theorems are found in Section 4. Theorem B is justified immediately and Theorem A is justified after establishing similar, but slightly more general statements for groups over fields of type (F) as in [\textit{J.-P. Serre}, Galois cohomology. Transl. from the French by Patrick Ion. Berlin: Springer (1997; Zbl 0902.12004)].\N\NIn addition, the authors provide an alternative proof in the characteristic \(0\) case of the following theorem from [\textit{M. S. Raghunathan} and \textit{A. Ramanathan}, Proc. Indian Acad. Sci., Math. Sci. 93, 137--145 (1984; Zbl 0587.14007)].\N\NTheorem: Let \(G\) be a connected reductive algebraic group over a field \(k\) and let \(\pi : B \to \mathbb{A}_k^1\) be a \(G\)-torsor over the affine line \(\mathbb{A}_k^1=\mathrm{Spec}(k[t])\). If \(\pi\) is trivialized by the base change from \(k\) to \(\overline{k}\) (separable closure), then \(\pi\) is obtained by the base change \(\mathbb{A}_k^1 \to \mathrm{Spec}(k)\) from a \(G\)-torsor \(\pi_0: B_0 \to \mathrm{Spec}(k)\).\N\NTheir argument uses an application of the fixed point theorem to show that any \(1\)-cocycle representing \(\pi: B \to \mathbb{A}_k^1\) must take values in the stabilizer of a closed simplex in the Bruhat-Tits building. Then, they apply results obtained earlier in the paper which describe such a subgroup to deduce that the cocycle is equivalent to a cocycle which only takes values in an underlying field, i.e., it does not use \(t\) and therefore comes from a torsor over \(\mathrm{Spec}(k)\) as claimed.