论文标题
在Anosov亚组下的二次形式的生长
Growth of quadratic forms under Anosov subgroups
论文作者
论文摘要
令$ \ mathbb {k} $等于$ \ Mathbb {k} $等于$ \ m athbb {r} $或$ \ mathbb {c} $。令$ o $为$ \ mathbb {k}^d $（其中$ 0 <p <d）$上的签名$（p，d-p）$的一种形式。令$ \ mathsf {s}^o $为$ pso（o）$的Riemannian对称空间的相应地理副本，内部的Riemannian对称空间$ PSL_D（\ Mathbb {k}）$。对于$ o $的某些选择，每$ t $都足够大，我们显示了$ \ \ mathsf {s}^o $和$ργ\ cdot \ cdot \ mathsf {s}^o $之间的距离的$γ\inγ$的指数界限。在一个额外的假设下，例如，当连接$γ$的边界时，我们显示了相对于双重极限锥体内部功能的计数函数的渐近性为$ t \ rightarrow \ infty $。
Let $ρ:Γ\rightarrow PSL_d(\mathbb{K})$ be a Zariski dense Borel-Anosov representation, for $\mathbb{K}$ equal to $\mathbb{R}$ or $\mathbb{C}$. Let $o$ be a form of signature $(p,d-p)$ on $\mathbb{K}^d$ (where $0<p<d)$. Let $\mathsf{S}^o$ be the corresponding geodesic copy of the Riemannian symmetric space of $PSO(o)$, inside the Riemannian symmetric space of $PSL_d(\mathbb{K})$. For certain choices of $o$ and every $t$ large enough, we show exponential bounds for the number of $γ\inΓ$ for which the distance between $\mathsf{S}^o$ and $ργ\cdot\mathsf{S}^o$ is smaller than $t$. Under an extra assumption, satisfied for instance when the boundary of $Γ$ is connected, we show an asymptotic as $t\rightarrow\infty$ for the counting function relative to a functional in the interior of the dual limit cone.
