Symbolic approach and induction in the Heisenberg group

TL;DR

This paper explores the relationship between hyperbolic automorphisms of free groups and their associated flows in the Heisenberg group, revealing conjugacy to niltranslations with self-induced properties.

Contribution

It introduces a symbolic approach linking automorphisms in free groups to flows in the Heisenberg group, highlighting new connections and properties.

Findings

Homomorphisms in the Heisenberg group are associated with hyperbolic automorphisms.

First return-times of certain flows are conjugate to self-induced niltranslations.

The approach provides new insights into the structure of automorphisms and flows in the Heisenberg group.

Abstract

We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in "good" sections, are conjugate to niltranslations, which have the property of being self-induced.