Quasi-isometry of pairs: surfaces in graph manifolds

TL;DR

This paper demonstrates that in certain graph manifolds, infinitely many non-separable, horizontal surfaces have fundamental groups that cannot be matched by quasi-isometries of the ambient manifold's fundamental group, highlighting complex geometric relationships.

Contribution

It constructs explicit examples of surfaces in a closed graph manifold with fundamental groups that are not quasi-isometric under any quasi-isometry of the ambient space.

Findings

Existence of a closed graph manifold with infinitely many non-separable, horizontal surfaces.

No quasi-isometry of the manifold's fundamental group can map these surfaces' groups to each other.

Shows limitations of quasi-isometric classification for surface subgroups 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 $n \neq m$.