Ratio Set of Boundary Actions - Zhou, Tianyi (Université d'Ottawa / University of Ottawa, 2023)
Given an action of a countable group with a quasi-invariant measure, there exists a multiplicative group in (0, ∞), called the ratio set of the group action, which in a sense describes the values of the Radon-Nikodym derivative. The main purpose of this thesis is to find the ratio set of the action of a finitely generated free group Ƒ on its topological boundary ∂Ƒ (the set of infinite words) for a certain natural class of quasi-invariant boundary measures. -- In Section 1, we focus on the general ergodic theory of equivalence relations. We outline the set-up, borrow from [1], [4] the definitions of the central notions of the theory, including counting measures (Proposition 1.8), quasi-invariance (Definition 1.6), Radon-Nikodym cocycle (Definition 1.15) and raio set (Definition 1.19), and illustrate them on the example of the orbit equivalence relation of a Markov shift (Definition 1.22). We also introduce the principal object: the boundary action of a finitely generated free group (see Section 1.2). -- In Section 2, we define the class of multiplicative Markov measures (Definition 2.1). These are the measures on a topological Markov chain entirely determined just by an initial (base) distribution and the admissibility matrix; the transition probabilities are then just the normalized restrictions of the base distribution onto the set of admissible transitions (see [7]). In the case of the free group, its boundary has a natural structure of a topological Markov chain (determined by the irreducibility condition from the definition of a free group: consecutive letters should not cancel each other), and in this case, we show that the multiplicative Markov measures are precisely the ones for which the Radon-Nikodym cocycle is a product cocycle (i.e. a cocycle whose potential only depends on the first letter of the input; see Definition 2.8). The final result of this section is an explicit description of the ratio set of the boundary action with respect to multiplicative Markov measures. -- In Section 3, given a probability measure 𝜇 on the set of free generators and their inverses, the definition of the associated nearest neighbor random walk is given. According to Furstenberg's Theorem (proof provided in Appendix), in this random walk, sample paths converge almost surely to a random boundary point, and the resulting limit distribution on the boundary of the free group is called the harmonic measure of the random walk (see Section 3.1). We show that the harmonic measure is a multiplicative measure (Theorem 3.3), and therefore the results of Section 2 allow us to describe the ratio set of the harmonic measure (Theorem 3.5). A significant role in these considerations is played by the passage probabilities of the random walk (given a group element, the probability that it is ever visited by a random walk). Since the harmonic measure is multiplicative, its potential only depends on the first letter, and this dependence actually amounts to taking the inverse of the corresponding passage probability (Proposition 2.9, Remark 2.10). Finally, we establish a one-to-one correspondence between three families of numbers indexed by the alphabet of the free group and subject to natural conditions; these are the step distributions of the random walk, the base of the harmonic measure (which is multiplicative Markov) and the family of passage probabilities (Theorem 3.6). -- In Section 4, we discuss another method for finding the ratio set of the harmonic measure based on using Martin theory (see [2]). -- In the Appendix, we prove Furstenberg's Theorem, a result used for defining the harmonic measure in Section 3. Actually, it is applicable not only for the nearest neighbor random walk (i.e. not only when the probability measure 𝜇 is supported on the alphabet set) but also the more general case where the support of the step distribution generates the free group. Moreover, in addition to the existence it also characterizes the harmonic measure as the unique 𝜇-stationary measure on the boundary
Keywords: Free group Gromov boundary; Free group action; Ratio set; Radon-Nikodym cocycle; Markov shift; Furstenberg's Theorem