Poisson boundary of groups acting on real trees

2009/11/03 by François Gautero, Gautero, François, Frédéric Mathéus +1
Mathematics · #20F65 #20F67 #20P05 #60B15 #60J50 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Probability (math.PR) #math.GT #math.PR #msc:20F65 #msc:20F67 #msc:20P05 #msc:60B15 #msc:60J50

paper · pdf · doi:10.48550/arxiv.0911.0616

42 pages. This is a majorly revised version. The introduction is completely rewritten, the statements are simplified. A new section entitled "The method" is added just after the introduction. More details are given on the case of a direct product of two finitely generated free groups

openalex publication_date 2009/11/03 · arxiv created 2011/01/07 · arxiv updated 2011/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic groups. We rely upon the description of Poisson boundaries by means of a topological compactification as developed by Kaimanovich. All the groups studied here share the property of admitting a sufficiently complicated action on some real tree.

Citations

• The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth
• Cannon-Thurston maps for hyperbolic group extensions
• Geodesics in trees of hyperbolic and relatively hyperbolic groups

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• Random walks on products of trees and certain t.d.l.c. groups
• The Poisson formula for groups with hyperbolic properties
• Poisson--Furstenberg boundaries of fundamental groups of closed 3-manifolds
• A construction of the measurable Poisson boundary: from discrete to continuous groups