Ergodic properties of boundary actions and the Nielsen–Schreier theory

We study the basic ergodic properties (ergodicity and conservativity) of the action of an arbitrary subgroup H of a free group F on the boundary ∂F with respect to the uniform measure. Our approach is geometrical and combinatorial, and it is based on choosing a system of Nielsen–Schreier generators...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2012-06, Vol.230 (3), p.1340-1380
Main Authors: Grigorchuk, Rostislav, Kaimanovich, Vadim A., Nagnibeda, Tatiana
Format: Article
Language:English
Subjects:
Boundary action
Free group
Hopf decomposition
Horospheric limit set
Nielsen–Schreier theory
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Summary:We study the basic ergodic properties (ergodicity and conservativity) of the action of an arbitrary subgroup H of a free group F on the boundary ∂F with respect to the uniform measure. Our approach is geometrical and combinatorial, and it is based on choosing a system of Nielsen–Schreier generators in H associated with a geodesic spanning tree in the Schreier graph X=H∖F. We give several (mod 0) equivalent descriptions of the Hopf decomposition of the boundary into the conservative and the dissipative parts. Further, we relate conservativity and dissipativity of the action with the growth of the Schreier graph X and of the subgroup H (≡ cogrowth of X), respectively. We also construct numerous examples illustrating connections between various relevant notions.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2012.03.013