If $(X,d)$ is a complete, locally-compact path metric space, then any two points can be joined by a minimizing geodesic (using the Arzela-Ascoli and lowersemicontinuity of length). Finite-dimensional complete Alexandrov spaces are locally compact.

I look forward to reading your paper (and of course seeing what you et al. have to say about homological stability for $SL_n Z$). While I'm sure your paper will clarify, what homology are you considering on the quotient $T_n / SL_n \mathcal{O}$? Are you considering a cellular homology (with $Z$ coefficients) on the naive quotient, or is your theorem stating surjectivity in rational homology (i.e. finite reducible subgroups in $SL_n \mathcal{O}$ prevent the action on $T_n$ from being free).

I find your comment informative, thanks. However I do not see (and don't believe) the existence of a $G_Z$-equivariant cocompact retract forcing equivariant contractibility at-infinity. Not even for a finite-index torsion free (and possibly neat) $\Gamma < G_Z$. What argument do you have in mind for this point?

Useful rephrasing.