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} o N}_{n i n m a t h b b N}

such that there does not exist a quasi-isometry

p i_{1} (N) o p i_{1} (N)

taking

p i_{1} (S_{n})

to

p i_{1} (S_{m})

within a finite Hausdorff distance when

n e q m

.

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