Whitehead moves for G-trees

TL;DR

This paper extends Whitehead moves to G-trees, demonstrating that two fundamental moves can relate any two reduced G-trees within the same deformation space, with applications in graph of groups theory.

Contribution

It introduces two Whitehead moves for G-trees that connect all reduced trees in a deformation space, with factorizations into simpler moves.

Findings

Two moves suffice to relate any two reduced G-trees

Moves can be factored into three simpler moves

Applications in graphs of groups theory

Abstract

We generalize the familiar notion of a Whitehead move from Culler and Vogtmann's Outer space to the setting of deformation spaces of G-trees. Specifically, we show that there are two moves, each of which transforms a reduced G-tree into another reduced G-tree, that suffice to relate any two reduced trees in the same deformation space. These two moves further factor into three moves between reduced trees that have simple descriptions in terms of graphs of groups. This result has several applications.