The simple loop conjecture for $3$–manifolds modeled on Sol

Abstract

The simple loop conjecture for $3$–manifolds states that every $2$–sided immersion of a closed surface into a $3$–manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the loop theorem to immersed surfaces. We prove the conjecture in the case that the target $3$–manifold admits a geometric structure modeled on $Sol$.