Let $X$ be a (possibly non-compact) complete Riemannian manifold on which a Lie group $G$ acts smoothly and properly. Fix $x_0\in X$, and let $\rho(x)$ denote the Riemannian distance function from $x\in X$ to $x_0$.
I wish to show that, for a large enough $\kappa\in\mathbb{R}$, the following function is in $L^1(G)$:
$$g\mapsto ||e^{-\kappa\rho(x)-\kappa\rho(g^{-1}x)}||_\infty.$$
Here the supremum norm is taken over $x\in X$.
Many thanks.
Edit: my question is simply: how might one go about showing this function is in $L^1(G)$?