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