Corporate Finance: From External Financing Needed(EFN) to Sustainable Growth Rate
We can replace growth rate in the formulation as below
$$\frac{A_0}{S_0}gS_0-\frac{L_0}{S_0}gS_0-PM(1+g)S_0b=0$$
$$(A_0-L_0)g-PMS_0b-PMS_0bg=0$$
$$g=\frac{bPMS_0}{A_0-L_0-bPMS_0}$$

In the paper you linked the author assumed $A_0$ and $L_0$ to be total assets and total debt (pg.11); therefore $A_0-L_0=E_0$
$$g=\frac{bPMS_0}{E_0-bPMS_0}$$
$$g=\frac{b\frac{PMS_0}{E_0}}{1-b\frac{PMS_0}{E_0}}$$
$$g=\frac{b\times ROE}{1-b\times ROE}$$