Length Function Compatibility for Group Actions on Real Trees

TL;DR

This paper explores when the sum of length functions from irreducible group actions on real trees also corresponds to an irreducible action, providing geometric and algebraic characterizations that generalize existing algorithms.

Contribution

It introduces a geometric characterization for the existence of a common refinement of group actions on real trees, extending prior algorithmic approaches.

Findings

Provides a geometric criterion for common refinement existence.

Generalizes Behrstock, Bestvina, and Clay's algorithm to broader settings.

Establishes algebraic characterizations via translation length functions.

Abstract

Let $G$ be a finitely generated group. Given two length functions $\ell$ and $m$ of irreducible $G$ actions on real trees $A$ and $B$, when is the point-wise sum $\ell + m$ again the length function of an irreducible $G$ action on a real tree? Guirardel and Levitt showed that additivity is equivalent to the existence of a common refinement of $A$ and $B$, this equivalence is established using Guirardel's core. Moreover, in this case the sum $\ell + m$ is the length function of the common refinement of $A$ and $B$ given explicitly by the Guirardel core. The core can be difficult to compute in general. Behrstock, Bestvina, and Clay give an algorithm for computing the core for free group actions on simplicial trees. In this article we give a geometric characterization of existence of a common refinement that generalizes the criterion underlying Behrstock, Bestvina, and Clay's algorithm, as well as two equivalent characterizations in terms of the associated translation length functions.