An algebraic approach to certain cases of Thurston rigidity

TL;DR

This paper provides an algebraic proof of the transversality of certain intersection curves in the moduli space of degree 3 polynomials, using 3-adic integrality, and extends results to preperiodic critical points.

Contribution

It offers a purely algebraic proof of intersection transversality in polynomial moduli space and extends the results to cases with preperiodic critical points.

Findings

Proves 3-adic integrality of intersection points.

Establishes transversality of critical point loci.

Extends results to preperiodic critical points with tail length 1.

Abstract

In the moduli space of polynomials of degree 3 with marked critical points c_1 and c_2, let C_{1,n} be the locus of maps for which c_1 has period n and let C_{2,m} be the locus of maps for which c_2 has period m. A consequence of Thurston's rigidity theorem is that the curves C_{1,n} and C_{2,m} intersect transversally. We give a purely algebraic proof that the intersection points are 3-adically integral and use this to prove transversality. We also prove an analogous result when c_1 or c_2 or both are taken to be preperiodic with tail length exactly 1.