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Mar 4, 2018 at 4:33 history edited Ian Agol CC BY-SA 3.0
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Mar 3, 2018 at 10:29 vote accept aglearner
Mar 3, 2018 at 10:27 vote accept aglearner
Mar 3, 2018 at 10:29
Mar 3, 2018 at 1:42 comment added Ian Agol @aglearner: I think this might be more difficult (or rather, with worse bounds). Off the top of my head, I see how to get a bound which is of the form $dg^{Cg}$ for some constant $d$, which is superexponential. Extend the standard collection of loops to a triangulation with one vertex. Then one can change it to another triangulation with at most $C g\log g$ flips. Each flip changes the length by at most a factor of 2. See theorem 1.4 of this paper: arxiv.org/abs/1411.4285. I suspect one should be able to obtain better bounds though, maybe exponential (but I'm not conjecturing that)?
Mar 2, 2018 at 20:56 comment added aglearner Thank you! It looks indeed that this works. When I was asking the question I had in mind that this standard collection is the one from the very standard picture i.e. in $\pi_1(S,x)$ we have $[\gamma_1,\gamma_2]\cdot ...=1$ (like here: math.stackexchange.com/questions/479371/…). I wonder if your argument can be modified to give a good bound for this more restricted type of standard collections as well.
Mar 2, 2018 at 19:10 history answered Ian Agol CC BY-SA 3.0