2004/09/16 by Ilya Kapovich, Kapovich, Ilya, Gilbert Levitt +6
Mathematics · #57M05 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Primary 20F36 #Secondary 20E36 #math.GR #math.GT #msc:20E36 #msc:20F36 #msc:57M05
paper · pdf · doi:10.48550/arxiv.math/0409284
revised version, to appear in Transact. Amer. Math. Soc.; two .eps figures
openalex publication_date 2004/09/16 · arxiv created 2005/01/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when two elements g,h in a free group F have the property that for every free isometric action of F on an ℝ-tree X the translation lengths of g and h on X are equal. We give a combinatorial characterization of this phenomenon, called translation equivalence, in terms of Whitehead graphs and exhibit two difference sources of it. The first source of translation equivalence comes from representation theory and SL2 trace identities. The second source comes from geometric properties of groups acting on real trees and a certain power redistribution trick. We also analyze to what extent these are applicable to the tree actions of surface groups that occur in the Thurston compactification of the Teichmuller space.