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Texas A&M University
Mathematics

Events for 12/13/2023 from all calendars

Groups and Dynamics Seminar

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Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Zoran Sunik, Hofstra University

Title: On the Schreier spectra of iterated monodromy groups of critically-fixed polynomials

Abstract: Every self-similar group G of d-ary tree automorphisms induces a sequence of finite Schreier graphs X_n of the action of G on the level n of the tree, along with a sequence of d-to-1 coverings X_{n+1} -> X_n. There are interesting examples of self-similar groups for which the spectra of the corresponding Schreier graphs are described by backward iterations of polynomials of degree 2 (the first Grigorchuk group, the Hanoi Towers group, the IMG of the first Julia set, ...). In this talk, for every r>1, we provide examples of self-similar groups for which the spectra of the Schreier graphs are described by backward iterations of polynomials of degree r. The examples come from the world of iterated monodromy groups of critically-fixed polynomials. A critically-fixed polynomial is a polynomial that fixes all of its critical points. In general, if we start with any post-critically finite polynomial f of degree d on the Riemann sphere, the iterated monodromy group of f (due to Nekrashevych) is a self-similar group acting on the d-ary rooted tree by automorphisms in such a way that the corresponding sequence of Schreier graphs approximates the Julia set of f and the coverings approximate the action of f on the Julia set. In our examples, the degree r of the polynomial that describes the spectra of the Schreier graphs coincides with the maximal local degree of f at the critical points.