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First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a topological disk.

Question. Let $(\Sigma,x)$ is a compact hyperbolic surface (without boundary) of genus $g$ and diameter $d$ with a marked point $x$. How to estimate from above in terms of $g$ and $d$ the number $A$ such that there exists a standard collection of geodesic loops on $\Sigma$ based at $x$, all shorter than $A$?

I would be especially grateful for a reference.

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I think that $A=2d$ will work, basically by applying Morse theory to the distance function from $x$. Morse theory for distance functions was originally considered by Gromov (and then Cheeger). For the case of surfaces, see a paper of Gershkovich (also see Gershkovich-Rubinstein for a Morse theory of distance functions).

Imagine taking metric balls $B_r(x)$ of radius $r, 0\leq r\leq d$ about $x$, so that $B_d(x)=S$. These will be the image of a hyperbolic disk $D_r$ at $\tilde{x}$ of radius $r$ in the universal cover $\mathbb{H}^2 \overset{\rho}{\to} S$. For $r$ sufficiently close to 0 ($\leq$ the injectivity radius at $x$), $B_r(x)$ will be an embedded image of the disk $\rho(D_r)$. When $r=injrad_x(S)$, there will be some tangencies of pairs of points in $\rho(\partial D_r(x))$. Taking geodesics in $D_r$ from $\tilde{x}$ to these pairs of points in $\partial D_r(x)$, and projecting by $\rho $ to $S$, we see a collection of loops of length $2r$ based at $x$. Now, continue to let $r$ increase. Each time we see a pair of tangencies between pairs of points in $\partial B_r(x)$, create a "dual" loop. Once one reaches $r=d$, we will obtain a collection of loops cutting $S$ up into disks (one may argue this via Morse theory: self-tangencies are the only way to create 1-handles). Then taking a subset of $2g$ of these loops, one obtains a standard collection, each with length $\leq 2d$.

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  • $\begingroup$ 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. $\endgroup$
    – aglearner
    Commented Mar 2, 2018 at 20:56
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    $\begingroup$ @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)? $\endgroup$
    – Ian Agol
    Commented Mar 3, 2018 at 1:42

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