Ergodic properties of boundary actions and Nielsen--Schreier theory

Details Ergodic properties of boundary actions and Nielsen--Schreier theory Ergodic properties of boundary actions and Nielsen--Schreier theory

2009-01-29

arxiv.org/abs/0901.4734v3

Mathematics

Group Theory

minor editorial changes, added references; final version to appear in Advances in Mathematics

Scientific paper

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 $\partial F$ 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 $X=H\backslash F$. 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 $H$ ($\equiv$ cogrowth of $X$), respectively. We also construct numerous examples illustrating connections between various relevant notions.

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