It is unknown whether a hyperbolic group is residually finite. Is it known under the additional hypothesis of locally indicability? Namely: Is a locally indicable hyperbolic group, residually finite?
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The answer is "it is unknown". There are many potential counterexamples to the conjecture that all hyperbolic groups are residually finite. For example, let $\phi, \psi$ be two injective (but not surjective) emdomorphisms of the free group $F_2$. Consider the corresponding multiple ascending HNN extension of $F_2$, $G=\langle F_2,t,s| x^t=\phi(x), x^s=\psi(x), x\in F_2\rangle$. That group has a homomorphism onto the free group $\langle t,s\rangle$ whose kernel is locally free. So $G$ is locally indicable. It is an open problem whether $G$ is residually finite provided its presentation satisfies a small cancelation condition (see Problem 5.2 here ).