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 some assumptions on the translation lengths of its generators and their products. Our approach is combinatorial and relies solely on the defining axioms of pseudo-lengths, which are precisely the translation length functions for actions on $\Lambda$-trees. Furthermore, we show that, under some natural 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, discrete and free groups of isometries, and a description of the translation length functions arising from free actions on $\Lambda$-trees, where $\Lambda$ is Archimedean. This description is related to the Culler--Vogtmann outer space.