Geometric regularity of powers of two-dimensional squarefree monomial ideals

Let I be a two-dimensional squarefree monomial ideal of a polynomial ring S. We evaluate the geometric regularity, $$a_i$$-invariants of $$S/I^n$$ for $$i\ge 2$$. It turns out that they are all linear functions in n from $$n=2$$. Also, it is shown that $$\text{ g-reg }(S/I^n)={\text {reg}}(S/I^{(n)})$$ for all $$n\ge 1$$.