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Ergodic properties of boundary actions and Nielsen--Schreier theory

Kaimanovich, Vadim A.
Year 2009
Abstract We study the basic ergodic properties (ergodicity and conservativity) of the action of a subgroup $H$ of a free group $F$ on the boundary $\pt F$ 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 $X=H\bs F$. 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. On the other hand, our approach sheds a new light on entirely algebraic properties of subgroups of a free group. We also construct numerous examples illustrating the connections between various relevant notions.
Note minor editorial changes, added references
Stable URL http://archive-ouverte.unige.ch/unige:11877
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