Congruence counting in Schottky and continued fractions semigroups of $\operatorname{SO}(n, 1)$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914165009940480 |
|---|---|
| author | Sarkar, Pratyush |
| author_facet | Sarkar, Pratyush |
| contents | In this paper, the two settings we are concerned with are $Γ< \operatorname{SO}(n, 1)$ a Zariski dense Schottky semigroup and $Γ< \operatorname{SL}_2(\mathbb C)$ a Zariski dense continued fractions semigroup. In both settings, we prove a uniform asymptotic counting formula for the associated congruence subsemigroups, generalizing the work of Magee-Oh-Winter [arXiv:1601.03705] in $\operatorname{SL}_2(\mathbb R)$ to higher dimensions. Superficially, the proof requires two separate strategies: the expander machinery of Golsefidy-Varjú, based on the work of Bourgain-Gamburd-Sarnak, and Dolgopyat's method. However, there are several challenges in higher dimensions. Firstly, using the expander machinery requires a key input: the Zariski density and full trace field property of the return trajectory subgroups, newly introduced in [arXiv:2006.07787]. Secondly, we need to adapt Stoyanov's version of Dolgopyat's method to circumvent some technical issues while the main difficulty is to prove the key inputs: the local non-integrability condition (LNIC) and the non-concentration property (NCP). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_00545 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Congruence counting in Schottky and continued fractions semigroups of $\operatorname{SO}(n, 1)$ Sarkar, Pratyush Number Theory Dynamical Systems Spectral Theory 22E40, 37A44, 37C30 In this paper, the two settings we are concerned with are $Γ< \operatorname{SO}(n, 1)$ a Zariski dense Schottky semigroup and $Γ< \operatorname{SL}_2(\mathbb C)$ a Zariski dense continued fractions semigroup. In both settings, we prove a uniform asymptotic counting formula for the associated congruence subsemigroups, generalizing the work of Magee-Oh-Winter [arXiv:1601.03705] in $\operatorname{SL}_2(\mathbb R)$ to higher dimensions. Superficially, the proof requires two separate strategies: the expander machinery of Golsefidy-Varjú, based on the work of Bourgain-Gamburd-Sarnak, and Dolgopyat's method. However, there are several challenges in higher dimensions. Firstly, using the expander machinery requires a key input: the Zariski density and full trace field property of the return trajectory subgroups, newly introduced in [arXiv:2006.07787]. Secondly, we need to adapt Stoyanov's version of Dolgopyat's method to circumvent some technical issues while the main difficulty is to prove the key inputs: the local non-integrability condition (LNIC) and the non-concentration property (NCP). |
| title | Congruence counting in Schottky and continued fractions semigroups of $\operatorname{SO}(n, 1)$ |
| topic | Number Theory Dynamical Systems Spectral Theory 22E40, 37A44, 37C30 |
| url | https://arxiv.org/abs/2108.00545 |