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?



Locally Indicable Hyperbolic Groups
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](http://arxiv.org/pdf/0704.2899.pdf) ). 


