Taking stock

Our task was to cut G into tubes that grow more or less exponentially. To make sure that we have accomplished this, let us follow the course of a representative tube, as it threads its way down toward the plane. To start off, it drops straight down until it reaches height 1. Below height 1, its journey is divided into steps of three kinds.

• Steps spent over the residual piece of the plane. In the course of one of these steps, the tube remains vertical, and its hyperbolic horizontal cross-section grows exponentially.
• Steps spent over a sector or a triangle. In the course of one of these steps, the tube remains more or less vertical; its hyperbolic horizontal cross-section does not decrease, and increases by a definite factor from the beginning to the end of the step.
• Steps spent taking care of a baby hemisphere. These steps come in sets of three. In the course of one of these sets of three steps, the tubes remain more or less vertical; the hyperbolic cross-section does not decrease, and increases by a definite factor between the beginning of the first step and the end of the third step.

Taken together, these facts imply that we have indeed succeeded in cutting G into tubes that grow more or less exponentially, with rate bounded below and factor bounded above by universal constants. But as we remarked before, this implies that we can do the same for the fundamental region , so there is a universal lower bound for , and hence for . qed