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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.05223 |
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Table of 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.