Pullback invariants of Thurston maps

TL;DR

This paper explores various pullback invariants of Thurston maps, analyzing their relationships across different mathematical structures such as curves, mapping class groups, Teichmüller spaces, and moduli spaces.

Contribution

It systematically studies the connections between multiple invariants of Thurston maps, revealing harmonious relationships among their properties.

Findings

Identifies relationships between invariants across different mathematical frameworks

Establishes connections between pullback relations, linear operators, and endomorphisms

Provides a unified perspective on Thurston map invariants

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 $\R$-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.