§ Key intuition for hyperbolicity allows us to control word length
- Suppose we are interested to find a such that
- We can think of this as a case where is the base edge of a triangle, is the opposite vertex, and are the other two sides:
- If the space is euclidea, then and can be as long as they want to be while stays the same. The angles will become larger, and the angle will become smaller as the length increases.
- In a hyperbolic space, because the angle sum is less than 180, if we move too far away, the will "bend" to maintain the angle sum to be less than 180. But this means that we have distorted the ! There is a bound to how long we can make before we start distorting . This bound on is exponential in the length of .
- Alternatively, in a CAT-0 space, the base of the triangle cannot be too far away from the centroid. So we can't have be moved too far away, because if gets too large.