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The engulfing property for 3–manifolds

Graham A Niblo and Daniel T Wise

Geometry & Topology Monographs 1 (1998) 413–418
DOI: 10.2140/gtm.1998.1.413
 arXiv: math.GT/9810198
Abstract
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We show that there are Haken 3–manifolds whose fundamental groups do not satisfy the engulfing property. In particular one can construct a ${\pi }_{1}$–injective immersion of a surface into a graph manifold which does not factor through any proper finite cover of the 3–manifold.

Keywords
double coset decompositions, subgroup separability, 3–manifolds, engulfing property
Mathematical Subject Classification
Primary: 20E26
Secondary: 20F34, 57M05, 57M25