Title: Cherlin's conjecture for almost simple groups of Lie rank 1,
Abstract: We prove Cherlin's conjecture, concerning binary primitive permutation
groups, for those groups with socle isomorphic to $\mathrm{PSL}_2(q)$,
${^2\mathrm{B}_2}(q)$, ${^2\mathrm{G}_2}(q)$ or $\mathrm{PSU}_3(q)$. Our method
uses the notion of a "strongly non-binary action".