Compact surfaces satisfy UW1(Σ) ≤ UW1(tilde Σ), virtually cyclic polyhedra satisfy UW1(X) ≤ 6 UW1(tilde X), and any counterexample to the width question reduces to 2-complexes.
Bowditch, Bilipschitz triangulations of R iemannian manifolds
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Compact surfaces satisfy UW1(Σ) ≤ UW1(tilde Σ), virtually cyclic polyhedra satisfy UW1(X) ≤ 6 UW1(tilde X), and any counterexample to the width question reduces to 2-complexes.