Busemann-Selberg Functions and Completeness for Dirichlet-Selberg domains in $SL(n,\mathbb{R})/SO(n,\mathbb{R})$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909651046498304 |
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| author | Du, Yukun |
| author_facet | Du, Yukun |
| contents | We establish a general completeness criterion for Dirichlet-Selberg domains in the symmetric space $SL(n,\mathbb{R})/SO(n)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Busemann-Selberg Functions and Completeness for Dirichlet-Selberg domains in $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ Du, Yukun Group Theory 20F65 (Primary) 22E40, 53C35 (Secondary) We establish a general completeness criterion for Dirichlet-Selberg domains in the symmetric space $SL(n,\mathbb{R})/SO(n)$. 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. |
| title | Busemann-Selberg Functions and Completeness for Dirichlet-Selberg domains in $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ |
| topic | Group Theory 20F65 (Primary) 22E40, 53C35 (Secondary) |
| url | https://arxiv.org/abs/2412.06987 |