Trivialité des groupes de Whitehead réduits avec applications à l'approximation faible et l'approximation forte

We prove some triviality results for reduced Whitehead groups and reduced unitary Whitehead groups for division algebras over a henselian discrete valuation field whose residue field has virtual cohomological dimension or separable dimension $\le 2$. These results are applied to show strong approximation for isotropic absolutely almost simple simply connected groups of type A. In particular, such a group defined over the function field of a nonreal curve $C\mathrm{/}k$ satisfies strong approximation if the base field $k$ is a number field, a $p$-adic field, $\mathbb{C}(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$ or a two-variable function field over $\mathbb{R}$.