$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Crossposted on MSE: https://math.stackexchange.com/questions/4261809/volume-of-a-geodesic-ball-in-sln-son

Question: **What is the volume of a geodesic ball of radius $r$ in the symmetric space $\SL(n) / {\SO(n)}$?**

Context: Let $M = \SL(n) / {\SO(n)}$.  We can view $M$ as the Riemannian manifold of real $n\times n$ positive definite matrices with determinant 1 endowed with the so-called affine-invariant metric:
$$g_P(U, V) = \operatorname{trace}(P^{-1} U P^{-1} V)$$
for $P \in M$ and $U, V \in \operatorname{Sym}(n)$ (symmetric $n \times n$ matrices).

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Attempts/what I know:

(1) Theorem 2.4 of [Gual-Arnau and Naveira - Volume of tubes in noncompact symmetric spaces][1] (screenshot attached) provides a formula for the volume of a geodesic ball for any noncompact symmetric space.  However, I am not very familiar with symmetric space theory, so I am unable to parse what this formula says for $\SL(n) / {\SO(n)}$.
Example 3.4.6 of the lecture notes [Symmetric spaces
(Frühlingssemester 2018)][3] give a formula for the roots $\alpha_j$ used in Theorem 2.4, and it is well-known the rank is $q = n-1$.  However, I do not understand how to determine the constant $c$.  I have an idea what the domain $C_0$ might be (the intersection of a sphere and a Weyl chamber—expressions for the Weyl chambers can be found), but I am not confident if this is correct.

(2) When $n=2$, we know $\SL(n) / {\SO(n)}$ is isometric to the hyperbolic plane of curvature $-1/2$.  However, I am looking for a formula for general $n$.

(3) It is well-known that $\SL(n) / {\SO(n)}$ contains a totally geodesic submanifold isometric to a hyperbolic space.  This gives us a lower bound on the volume of a ball.  However, this is not sufficient for my purposes.

(4) $\SL(n) / {\SO(n)}$ has constant Ricci curvature, so we can use the [Bishop–Gromov inequality][4] to get an upper bound on the volume.  However, this is not sufficient for my purposes.

[![Theorem 2.4 from Gual-Arnau and Naveira article][2]][2]


  [1]: https://publi.math.unideb.hu/load_pdf.php?p=509
  [2]: https://i.sstatic.net/oHPx4.png
  [3]: https://metaphor.ethz.ch/x/2021/fs/401-3226-00L/lecture_notes/symsp.pdf
  [4]: https://en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality