Polyhedral products, flag complexes and monodromy representations

TL;DR

This paper introduces a polyhedral product-based framework that constructs faithful monodromy representations of various group products into automorphism groups of free groups and linear groups, advancing understanding of their algebraic structures.

Contribution

It develops a machinery using polyhedral products to produce faithful monodromy representations of complex group products into automorphism and linear groups, which was not previously established.

Findings

Constructs faithful representations into Aut(F_n) and Out(F_n)

Realizes representations into SL(n,Z) and GL(n,Z)

Provides a new geometric approach to group representations

Abstract

This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups $\rm Aut(F_n)$ and outer automorphisms of free groups $\rm Out(F_n)$, respectively, as well as faithful representations of products of finite groups into the linear groups $\rm SL(n,\mathbb Z)$ and $\rm GL(n,\mathbb Z)$. These faithful representations are realized as monodromy representations.