I am studying Bestvina and Mess's results on the boundary of hyperbolic groups [The Boundary of Negatively Curved Groups. Journal of the American Mathematical Society Vol. 4, No. 3 (Jul., 1991), pp. 469-481], but I am having trouble with the proof of Corollary 1.4, especially the final part (starting from "Assuming $dim_R\partial\Gamma=i\geq k$...").

Could anybody spell the proof out please? Or even point out a more detailed version of the proof?

Thank you so much!