using ideas from geometric group theory we provide a novel approach to green's conjecture on syzygies of canonical curves. via a strong vanishing result for koszul modules we deduce that a general canonical curve of genus $g$ satisfies green's conjecture when the characteristic is zero or at least $(g+2)/2$. our results are new in positive characteristic (and answer positively a conjecture of eisenbud and schreyer), whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by voisin. joint work with aprodu, papadima, raicu and weyman.