论文标题
光谱间隙和指数级混合在几何有限双曲线上
Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds
论文作者
论文摘要
令$ \ Mathcal {M} =γ\ BackSlash \ Mathbb {H}^{D+1} $为几何有限的双曲线歧管，关键指数超过$ d/2 $。我们获得了$ l^2（\ Mathrm {t}^1（\ Mathcal {M}）$中的矩阵系数的精确渐近扩展，该指数错误项基本上与laplace Operator for $ l^2（Mathiill per fate lax and fations fate laples gap and of spectral Gap and lax and per lax and per fation and and cap）的指数错误。结合Bourgain，Gamburd和Sarnak的工作以及Golsefidy和Varju在扩展器上的概括，这意味着均匀的指数混合，以$ \ MATHCAL {M MATHCAL {M} $的一致性覆盖$ \ Mathcal {M} $时$ \ mathrm {so}^{\ circ}（d+1,1）$。
Let $\mathcal{M}=Γ\backslash\mathbb{H}^{d+1}$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d/2$. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in $L^2(\mathrm{T}^1(\mathcal{M}))$, with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on $L^2(\mathcal{M})$ due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varju on expanders, this implies uniform exponential mixing for congruence covers of $\mathcal{M}$ when $Γ$ is a Zariski dense subgroup contained in an arithmetic subgroup of $\mathrm{SO}^{\circ}(d+1,1)$.
