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The simple loop conjecture for 3-manifolds modeled on Sol. (English) Zbl 1353.57021

The simple loop conjecture for 3-manifolds states that, if a 2-sided immersion of a closed surface into a closed 3-manifold induces a non-injective map of fundamental groups, then the kernel of this induced map contains an element represented by an essential closed loop (for an embedding, this follows from the loop theorem of Papakyriakopoulos). The simple loop conjecture is known to hold for Seifert fiber spaces [J. Hass, Proc. Am. Math. Soc. 99, 383–388 (1987; Zbl 0627.57008)] and graph manifolds [J. H. Rubinstein and S. Wang, Comment. Math. Helv. 73, No.4, 499–515 (1998; Zbl 0916.57001)]. In the present paper, the simple loop conjecture is proved for 3-manifolds modeled on the Sol-geomety. This is one of Thurston’s eight 3-dimensional geometries (where Sol stands for solvable); the closed 3-manifolds belonging to this geometry are finitely covered by torus bundles over \(S^1\) with hyperbolic monodromy (the torus bundles with parabolic monodromy instead give Nil-manifolds, those with elliptic monodromy Euclidean 3-manifolds, and these 3-manifolds are Seifert fibered). Also, a group-theoretic version of the simple loop conjecture is discussed in the present paper, by considering homomorphisms of closed surface groups to other classes of groups (in particular, the cases of finite groups and again of surface groups have been considered).

MSC:

57M99 General low-dimensional topology
57N10 Topology of general \(3\)-manifolds (MSC2010)
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