Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

δ is a geodesic.

      If δ is not a geodesic, then there is a small subarc δ′ ∈ δ such that δ′ is not a geodesic and δ′ lies in some disc Di with the halved radius.

      Let ε > 0 be the value by which the straightening process decreases the length of δ. Since γj converges to δ we have length(Δ(γj)) < length(γj) – ε/2 for sufficiently large j. On the other hand, length(γj) < length(δ)+ε/2. Then length(Δ(γj)) < length(γj) that gives a contradiction. Thus, any convergent subsequence of γj must converge to a geodesic.

      Assume that we can form a subsequence {γk} with all γk lying outside the neighbourhood U of the set of geodesics. The reasonings above show that there is a subsequence in {γk