# Monodromy representations and surfaces with maximal Albanese dimension

## TL;DR

This paper explores the connection between surfaces of general type with maximal Albanese dimension and monodromy representations of the braid group, providing new examples and computational results for small cases.

## Contribution

It establishes a link between surface geometry and braid group representations, and computes these representations for n up to 9, including new conjectural examples for n=4.

## Key findings

- Computed monodromy representations for n up to 9
- Recovered known surfaces with p_g=q=2 for n=2,3
- Proposed new examples for n=4

## Abstract

We relate the existence of some surfaces of general type and maximal Albanese dimension to the existence of some monodromy representations of the braid group $\mathsf{B}_2(C_2)$ in the symmetric group $\mathsf{S}_n$. Furthermore, we compute the number of such representations up to $n=9$, and we analyze the cases $n \in \{2, \, 3, \, 4\}$. For $n=2, \, 3$ we recover some surfaces with $p_g=q=2$ recently studied (with different methods) by the author and his collaborators, whereas for $n=4$ we obtain some conjecturally new examples.