I am reading the paper "Elliptic and transversally elliptic index theory from the viewpoint of KK-theory" by G. Kasparov, and I have been trying to understand the following argument, which appears on page 1336, starting at "End of proof of Theorem 5.8". I am mainly concerned with understanding the proof of the following claim:
Let $X$ be a (Riemannian) manifold and let $G$ be a Lie group acting on $X$ isometric, proper and cocompactly (the latter means $X/G$ is compact). Then, the volume of balls in $X$, as a function of the radius, grows at most exponentially.
Now, as it is explained in the paper, the properties of the action imply that all sectional curvatures of $X$ are uniformly bounded. I can see that, since $X/G$ is compact, sectional curvatures therein are uniformly bounded. By a lifting argument and with the aid of, for instance, O'Neill's formula for submersions, I can understand that sectional curvatures are uniformly bounded on horizontal sections over $X$. I don't see how to extend this argument to all sections over $X$.
The other link I cannot make is the one between the sectional curvature being uniformly bounded and the volume of balls as a function of the radius growing at most exponentially, so any help on that is also welcome.