The continuous gradability of the cut-point orders of R-trees

19 May 2021
Sam Adam-Day

An $\mathbb R$-tree is a metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying $\mathbb R$-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part of the talk, I answer this question in the negative: there is a branchwise-real tree order which is not continuously gradable. In the second part, I show that a branchwise-real tree order is continuously gradable if and only if every embedded well-stratified (i.e. set-theoretic) tree is $\mathbb R$-gradable. This tighter link with set theory is put to work in the third part answering a number of refinements of the main question, yielding several independence results.

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