论文标题
克莱尼人群体的圆环计数和自我加入
Torus counting and self-joinings of Kleinian groups
论文作者
论文摘要
对于任何$ d \ geq 1 $，我们获得了Tori的计数和等分分配结果，$ d $ d $二维的圆环包装，在自动加以的$γ_ρ<\ prod_ = 1}^d}^d \ prod_ = 1}^d \ m mathrm {pslm {psl} _2（psl} _2 _2（\ nathrm）$ a的组中$ d $ -tuple of cocompact表示$ρ=（ρ_1，\ cdots，ρ_d）$。更准确地说，如果$ \ MATHCAL P $是$γ_ρ$ -Admussible $ d $ - 维圆环包装，那么对于任何有限的子集$ e \ subset \ subset \ mathbb {c}^d $带有$ \ partial e $的任何有限的子集$ e \ subset \ mathbb {c}^d $，其中包含在适当的真实algebraic upvariety中s^{δ_{l^1}（ρ）}}} \ cdot \＃\ { λ_ρ）。$$这里$ 0<δ_{l^1}（ρ）\ le 2/\ sqrt d $是$γ_ρ$的关键指数。 （\ mathbb {c} \ cup \ {\ infty \}）^d $是$γ_ρ$的限制集，$ω__ρ$是$ \ mathbb {c}^d \capλ_ρ$上的本地有限的borel量度，可以明确描述。我们认为的可接受的圆环包装类别自然来自克莱恩群体的Teichmüller理论。我们的工作将Oh-Shah在圆形包装（即一维圆环包装）上的先前结果扩展到$ d $ -torus包装。
For any $d\geq 1$, we obtain counting and equidistribution results for tori with small volume for a class of $d$-dimensional torus packings, invariant under a self-joining $Γ_ρ<\prod_{i=1}^d\mathrm{PSL}_2(\mathbb{C})$ of a Kleinian group $Γ$ formed by a $d$-tuple of convex cocompact representations $ρ=(ρ_1, \cdots, ρ_d)$. More precisely, if $\mathcal P$ is a $Γ_ρ$-admissible $d$-dimensional torus packing, then for any bounded subset $E\subset \mathbb{C}^d$ with $\partial E$ contained in a proper real algebraic subvariety, we have $$\lim_{s\to 0} { s^{δ_{L^1}(ρ) }} \cdot \#\{T\in \mathcal{P}: \mathrm{Vol} (T)> s,\, T\cap E\neq \emptyset \}= c_{\mathcal P}\cdot ω_ρ (E\cap Λ_ρ).$$ Here $0<δ_{L^1}(ρ)\le 2/\sqrt d$ is the critical exponent of $Γ_ρ$ with respect to the $L^1$-metric on the product $\prod_{i=1}^d \mathbb{H}^3$, $Λ_ρ\subset (\mathbb{C}\cup\{\infty\})^d$ is the limit set of $Γ_ρ$, and $ω_ρ$ is a locally finite Borel measure on $\mathbb{C}^d\cap Λ_ρ$ which can be explicitly described. The class of admissible torus packings we consider arises naturally from the Teichmüller theory of Kleinian groups. Our work extends previous results of Oh-Shah on circle packings (i.e. one-dimensional torus packings) to $d$-torus packings.
