The structure of the group {\slshape\bfseries G}(\kern-.1emk\/[t\kern.1em]): Variations on a theme of Soulé

Abstract

Following Soulé’s ideas from 1979, we give a presentation of the abstract group $\text{ G}\left(k\left[t\right]\right)$ for any semisimple (connected) simply connected absolutely almost simple $k$-group $\text{ G}\left(k\left[t\right]\right)$. As an application, we give a description of $\text{ G}\left(k\left[t\right]\right)$ in terms of direct limits, and show that the Whitehead group and the naive group of connected components of $\text{ G}\left(k\left[t\right]\right)$ coincide.