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Non-triviality of the Poisson boundary of random walks on the group $H(\mathbb{Z})$ of Monod

Part of: Markov processes Probability theory on algebraic and topological structures Special aspects of infinite or finite groups Graph theory

Published online by Cambridge University Press:  04 November 2019

BOGDAN STANKOV Open the ORCID record for BOGDAN STANKOV [Opens in a new window]
BOGDAN STANKOV*
Affiliation:
Département de mathématiques et applications, École normale supérieure, CNRS, PSL Research University, 75005Paris, France email [email protected]
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Abstract

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$, we prove that either $H$ is solvable or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group $F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].

Keywords

Random walks on groupsPoisson boundarySchreier graphThompson’s group Fsolvable group

MSC classification

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 05C81: Random walks on graphs
Secondary: 20F65: Geometric group theory 60J05: Discrete-time Markov processes on general state spaces 60J50: Boundary theory 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 1160 - 1189
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Copyright
© Cambridge University Press, 2019

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