Points with finite orbits for trace maps

TL;DR

This paper investigates the action of automorphisms of free groups on a real coordinate space via trace maps, identifying all finite orbits which relate to finite subgroups of SL(2,C) and rational points on a torus.

Contribution

It characterizes all finite orbits of the trace map action, linking them to finite subgroups of SL(2,C) and rational points on a torus quotient.

Findings

Finite orbits correspond to finite subgroups of SL(2,C).

A dense set of rational points in a torus quotient also forms finite orbits.

The structure of these orbits is explicitly determined.

Abstract

We study an action of ${\rm Aut}(F_n)$ on $\mathbb{R}^{2^n-1}$ by trace maps, defined using the traces of $n$-tuples of matrices in $\mathrm{SL}(2,\mathbb{C})$ having real traces. We determine the finite orbits for this action. These orbits essentially come from (i) the finite subgroups of $\mathrm{SL}(2,\mathbb C)$, and (ii) a dense set of (rational) points in an embedded quotient of an $n$-torus.