I am trying to find a nice proof that a finitely presented group satisfying a linear isoperimetric inequality implies it is hyperbolic.
I came across these lecture notes, Theorem 3.22, but I am having a hard time following the proof:
https://www.math.ucdavis.edu/~kapovich/280-2020/hyplectures_papasoglu.pdf
I believe the star function stands for all the closed cells in the van Kampen diagram that touch that subcomplex of the van Kampen diagram.
Why would $l(L_1) \geq l’ - 2\rho$ ? I understand L is a geodesic, but I am unsure why the maximum length of the relators is in this inequality. I am also unsure why applying this star function $12K$ times gives that sequence, would we not be taking off $2\rho$ each time?
If anyone can make this proof more clear to me or provide different proof that would be appreciated.