Singular points for cone actions on the product of certain homogeneous spaces

Abstract

In this paper, we investigate divergent orbits for cone actions on products of certain homogeneous spaces. We introduce a notion of essential singularity for such actions, and estimate the Hausdorff dimension of the corresponding singular set. In particular, let $G/\Gamma =\mathrm{SL}(2,\mathbb{R}{)}^s/\mathrm{SL}(2,\mathbb{Z}{)}^s$, and let $C$ be a cone in the positive Weyl chamber with angular aperture $\epsilon >0$. Then the Hausdorff dimension of the set of points with essential divergent orbits under $C$ satisfies that when $\epsilon \in (0,\frac{1}{64})$, $$3s-\frac{1}{2}-4(s-1)\epsilon \le \dim D^e(C,G/\Gamma )\le 3s-\frac{1}{2}-\frac{1}{3}\epsilon .$$ This extends the previous result of An--Guan--Marnat--Shi \cite{AGMS} to higher-dimensional cone actions.