Ergodic properties of boundary actions and the Nielsen-Schreier theory

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• 作者：Rostislav Grigorchuk^a ; ^{grigorch@math.tamu.edu} ; Vadim A. Kaimanovich^b ; ^{vkaimano@uottawa.ca} ; ^{vadim.kaimanovich@gmail.com} ; Tatiana Nagnibeda^c ; ^{tatiana.smirnova-nagnibeda@unige.ch}
• 关键词：Free group ; Nielsen&ndash ; Schreier theory ; Hopf decomposition ; Boundary action ; Horospheric limit set
• 刊名：Advances in Mathematics
• 出版年：2012
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：20 June, 2012
• 卷：230
• 期：3
• 页码：1340-1380
• 文件大小：409 K

摘要

We study the basic ergodic properties (ergodicity and conservativity) of the action of an arbitrary subgroup of a free group on the boundary 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 associated with a geodesic spanning tree in the Schreier graph . 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 and of the subgroup ( cogrowth of ), respectively. We also construct numerous examples illustrating connections between various relevant notions.