Establishes uniform spectral gap for congruence covers of geometrically finite thin subgroups of SO(n,1) with cusps in the range 1/2 < δ_Γ ≤ n-2 for n ≥ 3 by proving Zariski density and full trace field properties of return trajectory subgroups.
Dynamics and Rigidity through the Lens of Circles
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abstract
We report on recent developments in the dynamics and rigidity of infinite-volume homogeneous spaces, viewed through the lens of circles. By addressing four natural questions about circle packings, we highlight the interplay between dynamics, geometry, and rigidity that defines the emerging frontier of homogeneous dynamics.
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math.DS 1years
2026 1verdicts
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Generalization of Selberg's $3/16$ theorem for geometrically finite thin subgroups of $\operatorname{SO}(n, 1)$
Establishes uniform spectral gap for congruence covers of geometrically finite thin subgroups of SO(n,1) with cusps in the range 1/2 < δ_Γ ≤ n-2 for n ≥ 3 by proving Zariski density and full trace field properties of return trajectory subgroups.