Limit pretrees for free group automorphisms: existence

TL;DR

This paper introduces a new real pretree structure associated with free group automorphisms, capturing growth dynamics and extending metric tree theory without relying on topology.

Contribution

It constructs a real pretree with a rigid action that detects exponential growth of elements, extending the classical metric tree approach.

Findings

Real pretree associated with free group automorphisms has a rigid, non-nesting action.

Loxodromic elements are characterized by exponential length growth.

The construction generalizes metric trees using only pretree intervals, avoiding topology.

Abstract

To any free group automorphism, we associate a real pretree with several nice properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism of the real pretree that represents the free group automorphism. Finally and crucially, the loxodromic elements are exactly those whose (conjugacy class) length grows exponentially under iteration of the automorphism; thus, the action on the real pretree is able to detect the growth type of an element. This construction extends the theory of metric trees that has been used to study free group automorphisms. The new idea is that one can equivariantly blow up an isometric action on a real tree with respect to other real trees and get a rigid action on a treelike structure known as a real pretree. Topology plays no role in this construction as all the work is done in the language of pretrees (intervals).