Translation length formula for two-generated groups acting on trees
Kamil Orzechowski
TL;DR
The paper develops a purely combinatorial framework for translation lengths of two-generated groups acting on $\Lambda$-trees by introducing and exploiting pseudo-lengths. It proves an explicit formula $2\lVert w\rVert = (\sum_{i=1}^{n-1} \lVert x_i x_{i+1} \rVert) + \lVert x_n x_1 \rVert$ for cyclically reduced words when a ping-pong pair $(a,b)$ satisfies $|\lVert a\rVert-\lVert b\rVert|<\min\{\lVert ab\rVert,\lVert ab^{-1}\rVert\}$, and establishes existence and uniqueness results for a pseudo-length on $F(a,b)$ with prescribed lengths $\lVert a\rVert=\alpha$, $\lVert b\rVert=\beta$, $\lVert ab\rVert=\gamma$, and $\lVert ab^{-1}\rVert=\delta$ under precise additive conditions ($\gamma-\alpha-\beta,\delta-\alpha-\beta \in 2\Lambda$, and either $\gamma=\delta>\alpha+\beta$ or $\max\{\gamma,\delta\}=\alpha+\beta$). The results are constructive, with an algorithmic approach to reduce to a ping-pong pair (Algorithm 1) and a description of purely hyperbolic pseudo-lengths in the Archimedean setting, tying into outer space for rank two. The work yields applications to properly discontinuous and discrete free actions and provides a framework for classifying translation lengths arising from free actions on $\Lambda$-trees, especially when $\Lambda$ is Archimedean.
Abstract
We investigate translation length functions for two-generated groups acting by isometries on $Λ$-trees, where $Λ$ 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 $Λ$-trees. Furthermore, we show that, under some natural conditions on four elements $α, β, γ, δ\in Λ$, 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 $Λ$-trees, where $Λ$ is Archimedean. This description is related to the Culler--Vogtmann outer space.