Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\operatorname{SO}(n, 1)$
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866909231943254016 |
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| author | Sarkar, Pratyush |
| author_facet | Sarkar, Pratyush |
| contents | Let $Γ$ be a convex cocompact thin subgroup of an arithmetic lattice in $\operatorname{SO}(n, 1)$. We generalize Selberg's $\frac{3}{16}$ theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a uniform resonance-free half plane for the congruence covers of the hyperbolic manifold $Γ\backslash \mathbb H^n$. This extends the work of Oh-Winter who established the $n = 2$ case. The theorem follows from uniform spectral bounds for the congruence transfer operators with holonomy. We employ Sarkar-Winter's frame flow version of Dolgopyat's method uniformly over the congruence covers as well as Golsefidy-Varjú's generalization of Bourgain-Gamburd-Sarnak's expansion machinery by using the properties that the return trajectory subgroups are Zariski dense and have trace fields which coincide with that of $Γ$. These properties follow by proving that the return trajectory subgroups have finite index in $Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_07787 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\operatorname{SO}(n, 1)$ Sarkar, Pratyush Dynamical Systems Spectral Theory 37D40, 22E40, 37A25 Let $Γ$ be a convex cocompact thin subgroup of an arithmetic lattice in $\operatorname{SO}(n, 1)$. We generalize Selberg's $\frac{3}{16}$ theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a uniform resonance-free half plane for the congruence covers of the hyperbolic manifold $Γ\backslash \mathbb H^n$. This extends the work of Oh-Winter who established the $n = 2$ case. The theorem follows from uniform spectral bounds for the congruence transfer operators with holonomy. We employ Sarkar-Winter's frame flow version of Dolgopyat's method uniformly over the congruence covers as well as Golsefidy-Varjú's generalization of Bourgain-Gamburd-Sarnak's expansion machinery by using the properties that the return trajectory subgroups are Zariski dense and have trace fields which coincide with that of $Γ$. These properties follow by proving that the return trajectory subgroups have finite index in $Γ$. |
| title | Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\operatorname{SO}(n, 1)$ |
| topic | Dynamical Systems Spectral Theory 37D40, 22E40, 37A25 |
| url | https://arxiv.org/abs/2006.07787 |