On trace set of hyperbolic surfaces and a conjecture of Sarnak and Schmutz
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908600244371456 |
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| author | Hao, Yanlong |
| author_facet | Hao, Yanlong |
| contents | In this paper, we investigate the trace set of a Fuchsian lattice. There are two results of this paper: the first is that for a non-uniform lattice, we prove Scmutz's conjecture: the trace set of a Fuchsian lattice exhibits linear growth if and only if the lattice is arithmetic. Additionally, we show that for a fixed surface group of genus bigger than 2 and any positive number $ε$, te set of cocompact lattice embedding such that their growth rate of trace set exceeds $n^{2-ε}$ has positive Weil-Petersson volume. We also provide an asymptotic analysis of the volume of this set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On trace set of hyperbolic surfaces and a conjecture of Sarnak and Schmutz Hao, Yanlong Group Theory 11F06, 20H10, 22E40 In this paper, we investigate the trace set of a Fuchsian lattice. There are two results of this paper: the first is that for a non-uniform lattice, we prove Scmutz's conjecture: the trace set of a Fuchsian lattice exhibits linear growth if and only if the lattice is arithmetic. Additionally, we show that for a fixed surface group of genus bigger than 2 and any positive number $ε$, te set of cocompact lattice embedding such that their growth rate of trace set exceeds $n^{2-ε}$ has positive Weil-Petersson volume. We also provide an asymptotic analysis of the volume of this set. |
| title | On trace set of hyperbolic surfaces and a conjecture of Sarnak and Schmutz |
| topic | Group Theory 11F06, 20H10, 22E40 |
| url | https://arxiv.org/abs/2410.05223 |