Torus counting and self-joinings of Kleinian groups

Abstract

For any $d\ge 1$, we obtain counting and equidistribution results for tori with small volume for a class of $d$-dimensional torus packings, invariant under a self-joining ${\Gamma }_{\rho }<{\prod }_{i=1}^d{\mathrm{PSL}}_2(\mathbb{C})$ of a Kleinian group $\Gamma$ formed by a $d$-tuple of convex cocompact representations $\rho =({\rho }_1,\cdots ,{\rho }_d)$. More precisely, if $\mathcal{P}$ is a ${\Gamma }_{\rho }$-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 $$\underset{s\to 0}{\lim }{s}^{{\delta }_{L^1}(\rho )}\cdot \#\{T\in \mathcal{P}:\mathrm{Vol}(T)>s,T\cap E\ne \varnothing \}=c_{\mathcal{P}}\cdot {\omega }_{\rho }(E\cap {\Lambda }_{\rho }).$$ Here $0<{\delta }_{L^1}(\rho )\le 2/\sqrt{d}$ is the critical exponent of ${\Gamma }_{\rho }$ with respect to the $L^1$-metric on the product ${\prod }_{i=1}^d{\mathbb{H}}^3$, ${\Lambda }_{\rho }\subset (\mathbb{C}\cup \{\infty \}{)}^d$ is the limit set of ${\Gamma }_{\rho }$, and ${\omega }_{\rho }$ is a locally finite Borel measure on ${\mathbb{C}}^d\cap {\Lambda }_{\rho }$ 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.

36 pages, 2 figures, To appear in Crelle's journal

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