20.35 (2022)
Open(Well-known problem). Let $S$ be a connected orientable hyperbolic surface of finite type and complexity at least 2. Does its mapping class group $\text{MCG}(S)$ have a non-elementary hyperbolic quotient?
See also a closely related question 16.108 about braid groups.
Progress
An affirmative answer is known when $S$ is a closed genus-2 surface (https://arxiv.org/pdf/2005.00567.pdf), and in the same paper it is shown that the answer will be affirmative provided certain hyperbolic groups are residually finite.
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