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We study the subsets $V_k(A)$ of a complex abelian variety $A$ consisting inthe collection of points $x\in A$ such that the zero-cycle $\{x\}-\{0_A\}$ is$k$-nilpotent with respect to the Pontryagin product in the Chow group. Thesesets were introduced recently by Voisin and she showed that $\dim V_k(A) \leqk-1$ and $V_k(A)$ is countable for a very general abelian variety of dimensionat least $2k-1$.  We study in particular the locus $\mathcal V_{g,2}$ in the moduli space ofabelian varieties of dimension $g$ with a fixed polarization, where $V_2(A)$ ispositive dimensional. We prove that an irreducible subvariety $\mathcal Y\subset \mathcal V_{g,2}$, $g\ge 3$, such that for a very general $y \in\mathcal Y $ there is a curve in $V_2(A_y)$ generating $A$ satisfies $\dim\mathcal Y \le 2g - 1.$ The hyperelliptic locus shows that this bound is sharp.

On the dimension of Voisin sets in the moduli space of abelian varieties