发表机构
Institute for Advanced Study; Simons Laufer Mathematical Sciences Institute; Yale University; University of Toronto（高等研究院; 西蒙斯劳弗数学科学研究所; 耶鲁大学; 多伦多大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对含抛物元的富克斯群的扎里斯基稠密正表示，证明了其商空间上Bowen–Margulis–Sullivan测度的定量局部混合，推导了相关渐近结果，还推广至满足外围根平衡条件的阿诺斯表示。
AI 中文摘要
设Γ是PSL₂(ℝ)的无挠、非初等、有限生成且含抛物元的富克斯群，ρ:Γ→G是到连通分裂半单实代数群的保型、扎里斯基稠密正表示。我们证明了ρ(Γ)\backslash G/M上Bowen–Margulis–Sullivan测度的定量局部混合，其中M是兼容极大紧子群中极大分裂环面的中心化子。对ρ(Γ)极限锥内部的每个u，沿exp(tu)的光滑紧支函数的相关性，经t^((rank_ℝ G -1)/2)归一化后，可在t⁻¹的整数次幂中得到完整渐近展开。我们还得到了横向位移的均匀高斯渐近，带有可积余项。由此推导出L²(ρ(Γ)\backslash G)上拟正则表示的矩阵系数渐近，以及黎曼球中的轨道计数渐近。更一般地，这些结论适用于满足外围根平衡条件的Γ的扎里斯基稠密、相对博雷尔阿诺斯表示，包括到秩不超过3的分裂单群的相对博雷尔阿诺斯表示，以及保型几何有限实双曲表示的自接合。为证明这些结果，我们在Furstenberg边界中ρ(Γ)的极限集上发展了内在尖点几何，并用其构造可数马尔可夫编码；建立了无界向量值返回上同环的指数矩界和均匀畸变估计；利用外围嘉当增长和扎里斯基密度，建立了谱分析所需的定量非可积性。
英文摘要
Let $Γ<\operatorname{PSL}_2(\mathbb R)$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $ρ:Γ\to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen--Margulis--Sullivan measures on $ρ(Γ)\backslash G/M$, where $M$ is the centralizer of a maximal split torus in a compatible maximal compact subgroup. For every $u$ in the interior of the limit cone of $ρ(Γ)$, correlations of smooth compactly supported functions along $\exp(tu)$, normalized by $t^{(\operatorname{rank}_{\mathbb R}G-1)/2}$, admit a complete asymptotic expansion in integer powers of $t^{-1}$. We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on $L^2(ρ(Γ)\backslash G)$ and for orbital counting in Riemannian balls. More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of $Γ$ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations. To prove these results, we develop an intrinsic cusp geometry on the limit set of $ρ(Γ)$ in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.
Comments122 pages
