Finite presentability of \(\text{SL}_1(D)\). (Q995369)

Let \(E\) be a local field of nonzero characteristic \(p\), and let \(D\) be a (finite-dimensional) central division \(E\)-algebra. It is known that the (multiplicative) group \(\text{SL}_1(D)\) of elements of \(D\) of reduced norm \(1\) (over \(E\)) has the following two properties: (i) \(\text{SL}_1(D)\) is compact and totally disconnected, or equivalently, it is profinite; (ii) \(\text{SL}_1(D)\) possesses an open subgroup that is a pro-\(p\)-group (these subgroups are necessarily of finite indices). The purpose of the paper under review is to prove that \(\text{SL}_1(D)\) is finitely presented as a profinite group. As explained by the author, his main result settles the last open case of a question posed by Y. Barnea. Thereby, it completes the proof of the theorem which states that if \(G\) is a connected, simply connected (absolutely almost) simple algebraic group defined over a local field \(F\) of arbitrary characteristic, then every compact open subgroup \(U\) of the group \(G(F)\) of \(F\)-rational points of \(G\) is finitely presented as a profinite group. The proof of the main result is rather involved and technical which does not allow us to discuss it here. Its starting point is the fact that finite presentability is a commensurability invariant; also, it is equivalent to the finiteness of the continuous cohomology group \(H^2(G,\mathbb{F}_p)\) of some open pro-\(p\)-subgroup \(G\subseteq\text{SL}_1(D)\) with coefficients in a field \(\mathbb{F}_p\) with \(p\) elements. This amounts to saying that \(G\) has finitely many equivalence classes of elementary extensions, and significantly, this means that the depths of all elementary covers of \(G\) are uniformly bounded. When \(G=\{g\in\text{SL}_1(D):v(g-1)>0\}\), where \(v\) is the standard valuation of \(D\), the paper shows that the depth of any elementary cover of \(G\) does not exceed \(100p^3d\) (\(d\) being the Schur index of \(D\)). The proof relies on certain relations between cohomology of pro-\(p\)-groups and associated graded Lie algebras.