Pullback invariants of Thurston maps

Abstract:

Associated to a Thurston map $f: S^2 \to S^2$ with postcritical set $P$ are several different invariants obtained via pullback: a relation $\mathcal {S}_{P} {\stackrel {f}{\longleftarrow }} \mathcal {S}_{P}$ on the set $\mathcal {S}_{P}$ of free homotopy classes of curves in $S^2\setminus P$, a linear operator $\lambda _f: \mathbb {R}[\mathcal {S}_{P}]\to \mathbb {R}[\mathcal {S}_{P}]$ on the free $\mathbb {R}$-module generated by $\mathcal {S}_{P}$, a virtual endomorphism $\phi _f: \mathrm {PMod}(S^2, P) \dashrightarrow \mathrm {PMod}(S^2, P)$ on the pure mapping class group, an analytic self-map $\sigma _f: \mathcal {T}(S^2, P) \to \mathcal {T}(S^2, P)$ of an associated Teichmüller space, and an analytic self-correspondence $X\circ Y^{-1}: \mathcal {M}(S^2, P) \rightrightarrows \mathcal {M}(S^2, P)$ of an associated moduli space. Viewing these associated maps as invariants of $f$, we investigate relationships between their properties.