Arithmetic Representation Growth of Virtually Free Groups

TL;DR

This paper develops new methods combining quiver representation theory and Hall algebra techniques to compute representation growth and E-polynomials of character varieties for virtually free groups over finite fields.

Contribution

It introduces a novel approach to counting representations of virtually free groups using quiver and Hall algebra methods, enabling explicit computations of E-polynomials.

Findings

Computed E-polynomials of character varieties for specific groups

Established a new framework for representation counting in virtually free groups

Applied methods to groups like PSL_2(Z) and SL_2(Z)

Abstract

We adapt methods from quiver representation theory and Hall algebra techniques to the counting of representations of virtually free groups over finite fields. This gives rise to the computation of the E-polynomials of $\mathbf{GL}_d(\mathbb{C})$-character varieties of virtually free groups. As examples we discuss the representation theory of $\mathbb{D}_\infty$ , $\mathbf{PSL}_2(\mathbb{Z})$ , $\mathbf{SL}_2(\mathbb{Z})$ , $\mathbf{GL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\mathbb{Z})$ .