The finite-volume conjecture for outer space

Let $X_n$ be outer space, let $\mathrm{Out}(F_n)$ act on it naturally, and let $ds^2$ and $ds^2_{\varepsilon}$ be the two Riemannian metrics defined in the previous section. The quotient $\mathrm{Out}(F_n)\backslash X_n$ is equipped with the corresponding quotient Riemannian volume. The finite-volume conjecture. The volume of $\mathrm{Out}(F_n)\backslash X_n$ with respect to either of the Riemannian metrics $ds^2$ and $ds^2_{\varepsilon}$ is finite. This conjecture is motivated by the finite-volume quotients arising for mapping class groups and arithmetic groups; the source gives no resolution.