Translation length formula for two-generated groups acting on trees

Abstract

We investigate translation length functions for two-generated groups acting by isometries on $\Lambda$-trees, where $\Lambda$ is a totally ordered abelian group. In this context, we provide an explicit formula for the translation length of any element of the group, under certain assumptions on the translation lengths of its generators and their products. Our approach is purely combinatorial and uses only the defining axioms of pseudo-lengths. As shown by Parry, pseudo-lengths coincide with the translation length functions for actions on $\Lambda$-trees. Furthermore, we prove that, under certain conditions on four elements $\alpha, \beta, \gamma, \delta \in \Lambda$, there exists a unique pseudo-length on the free group $F(a,b)$ assigning these values to $a$, $b$, $ab$, $ab^{-1}$, respectively. Applications include results on properly discontinuous actions and discrete free groups of isometries. We also develop an algorithmic approach to studying translation length functions arising from free actions on $\mathbb{R}$-trees. Based on this, we state a conjecture that would lead to a description of $\mathrm{Aut}{(F_2)}$-orbits in the Culler-Vogtmann outer space.