Let $S_g$ be the fundamental group of a genus $g$ surface. What are the possible infinite quotients of $S_g$? Are they all free?
2026-03-27 02:59:51.1774580391
Infinite quotients of surface groups
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Like any nonelementary hyperbolic group, $\pi_1(S_g)$ is an SQ-universal group, which means that every countable group can be embedded in some quotient of $\pi_1(S_g)$. So no, not every quotient of $\pi_1(S_g)$ is free.