Consider a noncompact symmetric space $X=G/K$, where $G$ is the isometry group acting on $X$ (semisimple, noncompact, connected, with finite center) and $K$ the stabilizer of a fixed point $x_0\in X$. Let $n=\dim X$. Then, for every $x\in X$, for a small radius $r>0$, the volume growth of a geodesic ball $B_X(x,r)$ is $|B_X(x,r)|\sim r^n$.
Now, for a discrete, torsion free subgroup $\Gamma$ of $G$, the quotient $M=\Gamma \backslash G/K$ is a locally symmetric space, with the structure of a Riemannian manifold. Can we claim that for every $\tilde{x}\in M$ and a sufficiently small radius $r>0$, $|B_M(\tilde{x},r)|\sim r^n$?
If $\Gamma$ is discrete, than $M$ will have the same dimension as $X$, so the volume of small balls will have the same exponent in $r$.