the Torelli group of $S_g$, and also indicate how our results can be extended to classify roots of finite products of powers of commuting Dehn twists. integrated

Roots and their induced partitions
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In this section we shall introduce some preliminary notions, which will be used in later sections.

\[defn:sgc\] Let ${\operatorname{{\mathcal{C}}}}$ be a multicurve in $S_g$, and let $N$ be a closed annular neighbourhood of ${\operatorname{{\mathcal{C}}}}$.

(i) We denote the surface $\overline{S_g\setminus N}$ by $\widehat{S_g({\operatorname{{\mathcal{C}}}})}$.

(ii) The closed orientable surface obtained from $\widehat{S_g({\operatorname{{\mathcal{C}}}})}$ by capping off its boundary components is denoted by $S_g({\operatorname{{\mathcal{C}}}})$.

(iii) If ${\operatorname{{\mathcal{C}}}}$ is a nonseparating multicurve, then $S_g({\operatorname{{\mathcal{C}}}})$ is a connected surface whose genus we denote by $g_{{\operatorname{{\mathcal{C}}}}}$.

\[defn:pseudo\_multicurve\] Recall that a multicurve ${\operatorname{{\mathcal{C}}}}$ in $S_g$ is said to be *pseudo-nonseparating* if $S_g({\operatorname{{\mathcal{C}}}})$ is disconnected, but $S_g({\operatorname{{\mathcal{C}}}}')$ is not disconnected for any proper submulticurve ${\operatorname{{\mathcal{C}}}}'\subset {\operatorname{{\mathcal{C}}}}$.

(i) If $|{\operatorname{{\mathcal{C}}}}| = k$, we write ${\operatorname{{\mathcal{C}}}}^{(k)}$ for such a multicurve. Note that ${\operatorname{{\mathcal{C}}}}^{(1)}$ is a single separating curve.

(ii) A disjoint union of $m$ copies of ${\operatorname{{\mathcal{C}}}}^{(k)}$ is denoted by ${\operatorname{{\mathcal{C}}}}^{(k)}(m)$.

(iii) For integers $g \geq 0$ and $m \geq 1$, we define ${\operatorname{{\mathbb S}}}_g(m)$ to be the disjoint union of $m$ copies $\{S_g^1, S_g^2, \ldots, S_g^m\}$ 