Uniform, rigid branchwise-real trees

TL;DR

This paper constructs specific types of branchwise-real trees that are both rigid and uniform, using iterative methods and generic colourings to achieve desired structural properties.

Contribution

It introduces novel constructions of rigid, uniform branchwise-real trees with various uniformity conditions, expanding understanding of their automorphism groups.

Findings

Existence of rigid branchwise-real trees with uniform branching degrees

Construction of trees with all points branching and same degree

Development of a generic colouring technique in ZFC for tree construction

Abstract

A branchwise-real tree is a partial order which is a tree and in which every branch is isomorphic to a real interval. I give constructions of such trees which are both rigid (i.e. without non-trivial order-automorphisms) and uniform (in two different senses). Specifically, I show that there is a rigid branchwise-real tree in which every branching point has the same degree, one in which every point is branching and of the same degree, and finally one in which every point is branching of the same degree and which admits no order-preserving function into the reals. Trees are grown iteratively in stages, and a key technique is the construction (in ZFC) of a family of colourings of $(0,\infty)$ which is 'sufficiently generic', using these colourings to determine how to proceed with the construction.