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 on the set of free homotopy classes of curves in $S^2-P$, a linear operator on the free $\mathbb{R}$-module generated by these homotopy classes of curves, a virtual endomorphism on the pure mapping class group, an analytic self-map of an associated Teichmueller space, and an analytic self-correspondence on an associated moduli space. Viewing all of these objects as invariants of $f$, we investigate harmonious relationships between their properties.

48 pages, 3 figures