Busemann-Selberg Functions and Completeness for Dirichlet-Selberg domains in $SL(n,\mathbb{R})/SO(n,\mathbb{R})$
Group Theory
2025-06-18 v4
Abstract
We establish a general completeness criterion for Dirichlet-Selberg domains in the symmetric space . 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 Poincar\'e's Algorithm always holds in specific cases.
Keywords
Cite
@article{arxiv.2412.06987,
title = {Busemann-Selberg Functions and Completeness for Dirichlet-Selberg domains in $SL(n,\mathbb{R})/SO(n,\mathbb{R})$},
author = {Yukun Du},
journal= {arXiv preprint arXiv:2412.06987},
year = {2025}
}