Irreducibility of eventually $2$-periodic curves in the moduli space of cubic polynomials

Abstract: Consider the moduli space, $\mathcal{M}{3},$ of cubic polynomials over $\mathbb{C}$, with a marked critical point. Let $\mathscr{S}{k,n}$ be the set of all points in $\mathcal{M}{3}$ for which the marked critical point is strictly $(k,n)$-preperiodic. Milnor conjectured that the affine algebraic curves $\mathscr{S}{k,n}$ are irreducible, for all $k \geq 0, n>0$. In this article, we show the irreducibility of eventually $2$-periodic curves, i.e. $\mathscr{S}{k,2},\; k\geq 0$ curves. We also note that the curves, $\mathscr{S}{k,2},\; k\geq 0$, exhibit a possible splitting-merging phenomenon that has not been observed in earlier studies of $\mathscr{S}{k,n}$ curves. Finally, using the irreducibility of $\mathscr{S}{k,2}$ curves, we give a new and short proof of Galois conjugacy of unicritical points lying on $\mathscr{S}_{k,2}$, for even natural number $k$.