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.



Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter
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](https://books.google.com/books?id=7APzQMYf0ZsC&lpg=PA117&ots=inThz0IEpp&dq=gershkovich%20singularity%20theory%20riemannian&pg=PA117#v=onepage&q=gershkovich%20singularity%20theory%20riemannian&f=false) (also see [Gershkovich-Rubinstein](https://mathscinet.ams.org/mathscinet-getitem?mr=1621571) 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 $\leq 2d$. 


