# Groups and Dynamics Seminar

## Fall 2023

Date: | November 1, 2023 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Volodymyr Nekrashevych, Texas A&M University |

Title: | Locally connected Smale spaces |

Abstract: | We will discuss Ruelle-Smale dynamical systems (hyperbolic topological dynamical systems), a.k.a. Smale spaces. They are topological generalizations of Anosov diffeomorphisms. One of the main open questions is characterization of the algebraic Ruelle-Smale spaces (i.e., covered by hyperbolic automorphisms of nilpotent Lie groups) in purely topological terms. For example, no non-algebraic examples of locally connected Smale spaces are known. We will discuss known results about this problem and its connection to group theory. |

Date: | November 8, 2023 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Zheng Kuang, Texas A&M University |

Title: | Growth of groups with finitely many incompressible elements |

Abstract: | I will define the class of groups of bounded type with finite cycles from tile inflations. Then I will show that if the set of incompressible elements of a group in this class is finite, then this group has subexponential growth with a bounded power in the exponent. Examples will be provided. |

Date: | November 29, 2023 |

Time: | 3:00pm |

Location: | BLOC 628 |

Speaker: | Nataliya Goncharuk, Texas A&M University |

Title: | Renormalization operators in circle dynamics |

Abstract: | Renormalization techniques were used in several breakthroughs in circle dynamics. I will give a short survey of related ideas and results. After that, I will focus on our recent results with M. Yampolsky, with applications to the smoothness of Arnold’s tongues. |

Date: | December 13, 2023 |

Time: | 3:00pm |

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. |