Abstract:Let $\Gamma$ be a geometrically finite thin subgroup of an arithmetic lattice $\Gamma_0 < G := \operatorname{SO}(n, 1)$ and consider the congruence covers of $\Gamma \backslash G$. In the breakthrough work of Bourgain-Gamburd-Sarnak, the expansion machinery was used to establish a uniform spectral gap in the setting $(G, \Gamma_0) = (\operatorname{SL}_2(\mathbb{R}), \operatorname{SL}_2(\mathbb{Z}))$ when the critical exponent satisfies $\delta_\Gamma > \frac{1}{2}$. The main applications are affine sieve for $\Gamma$-orbits and uniform resonance-free half-planes for the resolvent of the Laplacian. These results were generalized in subsequent works by Mohammadi-Oh, Oh-Winter, the author, and Edwards-Oh. Yet, the region $\delta_\Gamma \in \bigl(\frac{1}{2}, n - 2\bigr]$ for $n \geq 3$ remains to be treated when there are cusps. The purpose of this paper is to fill in this gap in the literature. The difficulty lies in working with a countably infinite coding due to the presence of cusps. In particular, we incorporate new tools to prove the Zariski density and full trace field properties of the return trajectory subgroups.
From: Pratyush Sarkar [view email]
[v1]
Wed, 17 Jun 2026 04:20:23 UTC (85 KB)
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