Suppose $X$ is a complete Riemannian manifold on which a Lie group $G$ acts properly.
Let $\rho(x)$ be the distance function from a chosen point $x_0\in X$ to our point $x\in X$. Suppose also that all sectional curvatures of $X$ are uniformly bounded, so that the volume of balls in $X$ (as a function of radius) grows at most exponentially.
Under these conditions, how would one prove that the following function $f: G\rightarrow\mathbb{R}$ defined by
$$g\mapsto \int_X e^{-c\rho(x)-c\rho(g^{-1}x)}\,dx,$$
is in $L^1(G)$ for $c\in\mathbb{R}$ sufficiently large?
I'm not sure how to use the constraint on the volume growth of balls, so it would be very helpful if someone could write a fairly detailed calculation showing how this is done. Many thanks!