Distortion of surfaces in 3-manifolds

Abstract

Let $g:S\looparrowright N$ be a properly immersed ${\pi }_1$--injective surface in a non-geometric $3$--manifold $N$. We compute the distortion of ${\pi }_1(S)$ in ${\pi }_1(N)$ and show that how it is related to separability of ${\pi }_1(S)$ in ${\pi }_1(N)$. The only possibility of the distortion is linear, quadratic, exponential, and double exponential.

33 pages, 2 figure. Clarifications are given in Section 4. Figure 1 is redrawn. Section 5 is rewritten for clarify and to fill a gap in the proof of the lower case (section 5.2). This paper has been accepted for publication in Journal of Topology

References in corpus (2)

• A characterization on separable subgroups of 3-manifold groups
• Distortion of surfaces in graph manifolds

Cited by in corpus (2)

• R-covered foliations and transverse pseudo-Anosov flows in atoroidal pieces
• Subgroup distortion of 3-manifold groups