BGU Probability and Ergodic Theory (PET) seminar Ical Atom

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TBAIn this lecture I will tell the story of a small, anonymous conjecture from an unpublished master’s thesis, that turned out to generalize (a slightly weaker version of) the famous Hanna Neumann conjecture (HNC), which challenged dozens of mathematicians for nearly 60 years. We will discuss the following 3 seemingly unrelated problems: 1. What is the maximal possible rank of the intersection of 2 finitely generated subgroups of a free group? 2. How complicated (in the sense of Euler characteristic) must a graph be in order to contain many copies of another graph? 3. If we substitute random permutations for the letters of some free words so that they generate a random permutation group, how many invariant subsets do we expect to get?

…and what if we replace permutations with random invertible matrices over a finite field? This last question gives rise to a q-analog of the HNC, closing a circle of ideas.

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We will begin with defining the concepts of harmonic measure and different notions of dimensions. We will then connect those notions with what is called multi-fractal spectrum. Next, we will discuss finer features of the relationship between those dimensions. Lastly, we will define the universal counterparts and discuss an approximation theorem, showing the importance of domains arising from multifractal formalism. This is a developing story, based on a joint work with I. Binder.

Let F be a free group and K a field. The free group algebra K[F] bears a strong resemblance to F, making it an excellent tool in the study of free groups. For example, by a theorem due to Cohn and Lewin, one-sided ideals in K[F] are free as K[F]-modules, analogously to the Nielsen-Schreier theorem. I will discuss this resemblance, along with other motivations for our interest in K[F] arising from the theory of word measures. I will then present a new algorithm for deciding if a given element is part of some basis of a given ideal, similarly to what Whitehead’s algorithm performs in free groups. Based on joint work with Danielle Ernst-West and Doron Puder.

In this talk, we will explore the structure of ends of Schreier graphs associated with stationary random subgroups (SRS). We begin with the classical Freudenthal-Hopf theorem on the possible number of ends of Cayley graphs and extend it to the setting of random subgroups. First, we establish the result for Schreier graphs of invariant random subgroups (IRS) before further generalizing it to stationary subgroups.

We will then introduce the concept of stationary actions, stationary random subgroups, and present the key result: For a group Γ with a symmetric generating set, the number of ends of the Schreier graph of an SRS is almost surely 0, 1, 2, or infinite.

Finally, we will outline the proof of this theorem, emphasizing the emergence of the “no-core” phenomenon in stationary actions, which is an interpretation of Kac’s lemma within the framework of stationary actions.

This work studies the relation between two graph parameters, rho and Lambda. For an undirected graph G, rho(G) is the growth rate of its universal covering tree, while Lambda(G) is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW).
It is well known that rho(G) >= Lambda(G) for all graphs, and that graphs with rho=Lambda exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length k NBRW is O(1) if rho = Lambda and Omega(k) if $rho > \Lambda. As a consequence we exhibit infinitely many non-trivial examples of graphs with rho = Lambda.

Joint work with Idan Eisner, Tel-Hai College