Products of pairs of Dehn twists and maximal real Lefschetz fibrations

TL;DR

This paper investigates factorizations of elements in the modular group into two Dehn twists and applies these results to show that all maximal real elliptic Lefschetz fibrations are algebraic.

Contribution

It provides new insights into the factorization problem in the modular group and establishes the algebraicity of maximal real elliptic Lefschetz fibrations.

Findings

Characterization of factorizations into two Dehn twists

Proof that maximal real elliptic Lefschetz fibrations are algebraic

Connections between modular group factorizations and Lefschetz fibrations

Abstract

We address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.