said to be a *reduction system* for $\sigma \in \Mod({\Sigma})$ if $\sigma$ fixes $\calA$.

-   An element $\sigma \in \Mod({\Sigma})$ is said to be *reducible* if it admits a *non-empty* reduction system $\calA$; it is said to be *irreducible* otherwise.

-   An element $\sigma \in \Mod({\Sigma})$ is said to be *periodic* if its order is finite; it is said to be *aperiodic* otherwise.

Nielsen studied $\Mod(T^2) \cong \SL(2,\Z)$ in terms of the dynamical properties of its action on the Teichmüller space $\Teich(T^2) \cong \Hyp^2$ and its boundary [@Nielsen:MCG], and he essentially showed that irreducible elements are precisely the *Anosov elements*, i.e. the elements that can be represented by *Anosov homeomorphisms* of $T^2$. The far-reaching generalization of Nielsen’s work, dealing with surfaces ${\Sigma}$ that admit complete hyperbolic metrics, was developed by Thurston [@Thurston:Surfaces]. Among other things, Thurston introduced the notion of *pseudo-Anosov homeomorphisms* of ${\Sigma}$ and showed that irreducible elements of $\Mod({\Sigma})$ are precisely the *pseudo-Anosov elements*, i.e. the elements that can be represented by pseudo-Anosov homeomorphisms of ${\Sigma}$.

The Canonical Reduction of Reducible Elements {#Can-Red: Elts}
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The notion of the the *reduction* of an element of $\Mod({\Sigma})$ appeared in Thurston’s theory, and it was further studied in [@Birman-Lubotzky-McCarthy:AbSol]. If 