Pseudo-Anosov mapping classes not arising from Penner's construction

TL;DR

This paper demonstrates that most pseudo-Anosov mapping classes do not originate from Penner's construction, by analyzing Galois conjugates of their stretch factors and resolving a conjecture of Penner.

Contribution

It proves that, except for a few cases, pseudo-Anosov classes cannot be obtained from Penner's construction, addressing a longstanding conjecture.

Findings

Galois conjugates of stretch factors lie off the unit circle

Most pseudo-Anosov classes are not from Penner's construction

Resolves Penner's conjecture for most surfaces

Abstract

We show that Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner's construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner's construction. This resolves a conjecture of Penner.