Quasi-isometry of pairs: surfaces in graph manifolds

Abstract

We show there exists a closed graph manifold $N$ and infinitely many non-separable, horizontal surfaces $\{S_n\to N{\}}_{n\in \mathbb{N}}$ such that there does not exist a quasi-isometry ${\pi }_1(N)\to {\pi }_1(N)$ taking ${\pi }_1(S_n)$ to ${\pi }_1(S_m)$ within a finite Hausdorff distance when $n\ne m$.

19 pages, 1 figure