Busemann-Selberg Functions and Completeness for Dirichlet-Selberg domains in $SL(n,\mathbb{R})\mathrm{/}SO(n,\mathbb{R})$

We establish a general completeness criterion for Dirichlet-Selberg domains in the symmetric space $SL(n,\mathbb{R})\mathrm{/}SO(n)$. By introducing and analyzing Busemann-Selberg functions - which extend classical Busemann functions and capture asymptotic behavior toward the Satake boundary - we show that every gluing manifold or orbifold produced by Dirichlet-Selberg domain is complete. This result parallels the well-known hyperbolic case and ensures that the key completeness condition in Poincaré’s Algorithm always holds in specific cases.