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

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 $G\mathrm{/}\mathrm{\Gamma }=\mathrm{S}\mathrm{L}(2,\mathbb{R}{)}^s\mathrm{/}\mathrm{S}\mathrm{L}(2,\mathbb{Z}{)}^s$, 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 $\epsilon \in (0,\frac{1}{64})$, $$3s-\frac{1}{2}-4(s-1)\epsilon \le \dim D^e(C,G\mathrm{/}\mathrm{\Gamma })\le 3s-\frac{1}{2}-\frac{1}{3}\epsilon \mathrm{.}$$ This extends the previous result of An–Guan–Marnat–Shi \cite{AGMS} to higher-dimensional cone actions.