Some results on maps that factor through a tree

TL;DR

This paper characterizes when maps from simply-connected quasiconvex metric spaces to trees or Euclidean planes factor through a tree, based on conditions involving winding numbers and H"older continuity.

Contribution

It provides necessary and sufficient conditions for maps to factor through a tree, including specific criteria for Euclidean and Heisenberg group targets based on H"older exponents.

Findings

Maps with H"older exponent > 1/2 to Euclidean plane are characterized by winding number integrals.

Maps to the Heisenberg group with H"older exponent > 2/3 factor through a tree.

A new criterion links H"older continuity and topological factorization through trees.

Abstract

We give a necessary and sufficient condition for a map defined on a simply-connected quasiconvex metric space to factor through a tree. In case the target is the Euclidean plane and the map is H\"older continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over the winding number function. This in particular shows that if the target is the Heisenberg group equipped with the Carnot-Carath\'eodory metric and the H\"older exponent of the map is bigger than 2/3, the map factors through a tree.