The paper studies fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov’s description we are able to prove that the only such morphisms are forgetful maps. This allows us to show that the automorphism group of ${\overline{M}}_{0,n}$ is the permutation group on $n$ elements as soon as $n\ge 5$.

We study the group $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ of tame automorphisms of a smooth affine $3$-dimensional quadric, which we can view as the underlying variety of ${\mathrm{SL}}_2\left(ℂ\right)$. We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $\mathrm{CAT}\left(0\right)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ is linearizable, and that $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ satisfies the Tits alternative.