Lefschetz pencils and finitely presented groups

TL;DR

This paper constructs explicit Lefschetz fibrations with prescribed fundamental groups using twisted substitutions, providing bounds on the genus of Lefschetz pencils for given finitely presented groups.

Contribution

It introduces a method to explicitly determine monodromy of Lefschetz fibrations with specified fundamental groups, expanding the toolkit for 4-manifold topology.

Findings

Explicit monodromy for Lefschetz fibrations with given fundamental groups

Upper bounds for genus of Lefschetz pencils with prescribed fundamental groups

Application of twisted substitutions to construct Lefschetz fibrations

Abstract

In this paper, given a finitely presented group $\Gamma$, we provide the explicit monodromy of a Lefschetz fibration with $(-1)$-sections whose total space has fundamental group $\Gamma$ by applying "twisted substitutions" to that of the Lefschetz fibration constructed by Cadavid and independently Korkmaz. Consequently, we obtain an upper bound for the minimum $g$ such that there exists a genus-$g$ Lefschetz pencil on a smooth 4-manifold whose fundamental group is isomorphic to $\Gamma$.