pseudo-Anosov elements are infinite irreducible subgroups.

Torsion-Free Finite-Index Subgroups {#Mod-m}
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Before we proceed, let us first consider a useful family of subgroups of $\Mod({\Sigma})$. For each integer $m \geq 3$, we consider the natural homomorphisms $$\Mod({\Sigma}) \rightarrow \Aut(H_1({\Sigma};\Z)) \rightarrow \Aut(H_1({\Sigma};\Z/m\Z))$$ and let $\Mod_{(m)}({\Sigma})$ be the kernel of this composition of homomorphisms. $\Mod_{(m)}({\Sigma})$ is clearly a finite-index normal subgroup of $\Mod({\Sigma})$, and a classical theorem of Serre [@Serre:Rigidite] says that $\Mod_{(m)}({\Sigma})$ is torsion-free. For each subgroup $\Gamma < \Mod({\Sigma})$, we set $\Gamma_{(m)}:=\Gamma \cap \Mod_{(m)}({\Sigma})$; it is a torsion-free finite-index normal subgroup of $\Gamma$.

Ivanov observed that elements of $\Mod_{(m)}({\Sigma})$ possess other useful properties. For every element $\sigma \in \Mod_{(m)}({\Sigma})$ and for every reduction system $\calA$ of $\sigma$, the following statements hold [@Ivanov:MCGBook §1.2]:

-   $\sigma$ takes each component of ${\Sigma}$ to itself, and each component of $\partial {\Sigma}$ to itself;

-   $\sigma$ takes each component of $\calA$ to itself with its orientation preserved;

-   hence, the reduction ${\operatorname{\rho}}_{\!\calA}(\sigma) \in \Mod({\Sigma}_{\calA})$ takes each component of ${\Sigma}_{\calA}$ to itself, and each component of $\partial {\Sigma}_{\calA}$ to itself.

It follows that $\sigma$ can be restricted to each component of ${\Sigma}$, and ${\operatorname{\rho}}_{\!\calA}(\sigma)$ can be restricted to each component of ${\Sigma}_{\calA}$. Such restrictions have 