Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol.

Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow M$ of a closed orientable surface $\Sigma$ into a closed 3-manifold $M$. If $F_\ast \colon\, \pi_1 \Sigma \rightarrow \pi_1 M$ is not injective then there is an essential simple closed curve in $\Sigma$ that represents an element in the kernel of $F$.

If $F$ were an embedding then this would follow from Papakyriakopoulos's Loop Theorem. "To be an embedding" is not an algebraic property, so the Simple Loop Conjecture is more of a `$ \pi_1$ to 3-manifolds' statement than the loop theorem. It allows us to replace non-$\pi_1$-injective immersions by immersions of lower genus surfaces by surgery paralleling passage to a normal subgroup; So it does translate from algebra to topology.

Joel Hass proved the conjecture for Seifert-fibered spaces using geometrical techniques in 1987, and Hyam Rubinstein and Shicheng Wang proved the conjecture in 1998 for non-trivial graph manifolds (not Sol).

In Kirby's problem list it also states that the Simple Loop Conjecture arises in trying to characterize 3-manifold groups among Poincaré duality groups, but I'm not sure what Kirby means. I also don't know what else it implies.

Question: What wonderful things would follow from the Simple Loop Conjecture if it were true? Beyond it being a "natural question" and beyond the abstract considerations brought above, what is the significance of this conjecture?

Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol.

Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow M$ of a closed orientable surface $\Sigma$ into a closed 3-manifold $M$. If $F_\ast \colon\, \pi_1 \Sigma \rightarrow \pi_1 M$ is not injective then there is an essential simple closed curve in $\Sigma$ that represents an element in the kernel of $F_\ast$.

If $F$ were an embedding then this would follow from Papakyriakopoulos's Loop Theorem. "To be an embedding" is not an algebraic property, so the Simple Loop Conjecture is more of a `$ \pi_1$ to 3-manifolds' statement than the loop theorem. It allows us to replace non-$\pi_1$-injective immersions by immersions of lower genus surfaces by surgery paralleling passage to a normal subgroup; So it does translate from algebra to topology.

Joel Hass proved the conjecture for Seifert-fibered spaces using geometrical techniques in 1987, and Hyam Rubinstein and Shicheng Wang proved the conjecture in 1998 for non-trivial graph manifolds (not Sol).

In Kirby's problem list it also states that the Simple Loop Conjecture arises in trying to characterize 3-manifold groups among Poincaré duality groups, but I'm not sure what Kirby means. I also don't know what else it implies.

Question: What wonderful things would follow from the Simple Loop Conjecture if it were true? Beyond it being a "natural question" and beyond the abstract considerations brought above, what is the significance of this conjecture?