The structure of the~congruence kernel for~$\mathrm{SL}_2$ in the~case of a~global field of positive characteristic

P. A. Zalesskii

Sbornik. Mathematics, Tome 77 (1994) no. 2, pp. 489-495

Résumé

In this paper the following description of the congruence kernel $C(\mathrm{SL}_2,\mathcal O)$ is given, where $\mathcal O$ is the coordinate ring of an affine curve obtained by removing a point from a projective curve over a finite field $k_0$. Theorem. {\it $C(\mathrm{SL}_2,\mathcal O)=(*_{x\in X}H_x)*P$ is the free profinite product over a separable space $X$ of groups $H_x$ that are isomorphic to the direct product $\prod\mathbb{Z}/p\mathbb{Z}$ of a continuum of groups of order $p=\operatorname{char}(k_0)$, and a separable projective group $P$ each of whose open subgroups is free. } The proof uses a general result on normal subgroups of free profinite products.