# Groups and Dynamics Seminar

## Fall 2017

Date: | September 13, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Robin Tucker-Drob, Texas A&M |

Title: | Invariant means and inner amenable groups |

Abstract: | An action of a group G on a set X is said to be amenable if X admits a G-invariant mean (i.e., finitely additive probability measure). The group G is said to be amenable if the left translation action of G on itself is an amenable action. While these notions were introduced in 1929 by von Neumann, a systematic study of amenable actions of nonamenable groups was not initiated until roughly 1990. I will discuss this setting, and how the tension which arises between the nonamenability of the group and the amenability of the action results in surprising structural consequences for the acting group. This tension becomes particularly pronounced in the case of an atomless mean for the conjugation action, i.e., when the group is inner amenable. I will highlight some recent results and applications of inner amenability, particularly to orbit equivalence and measured group theory. |

Date: | September 20, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Volodymyr Nekrashevych, Texas A&M |

Title: | Amenability of iterated monodromy groups for some complex rational functions |

Abstract: | It is an open question if iterated monodromy groups of complex rational functions are amenable. Another open question is if groups generated by automat of polynomial activity growth are amenable. We prove that if the iterated monodromy group of a complex rational function is generated by an automaton of polynomial activity growth, then the group is amenable. At first, we show that orbital Schreier graphs of iterated monodromy groups are recurrent, by comparing the random walk on the graph with the Brownian motion on the associated Riemanian surfaces, Then we prove amenability of the group using techniques of extensive amenability. This is a joint work with K. Pilgrim and D. Thurston. |

Date: | September 27, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Robin Tucker-Drob, Texas A&M |

Title: | Invariant means and inner amenable groups II |

Abstract: | I will present the proof of the characterization of inner amenable linear groups which I mentioned in my last talk: A linear group G is inner amenable if and only if there is an infinite amenable normal subgroup N of G such that G/C_G(N) is amenable. The main ingredient is a "forgotten lemma" of S.G. Dani from 1985 concerning amenable actions which satisfy a certain chain condition. Time permitted, I will also discuss a closely related characterization of linear groups which are J.S.-stable. |

Date: | October 11, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Rostislav Grigorchuk, Texas A&M |

Title: | On spectra of groups of intermediate growth |

Abstract: | I will explain how to compute the spectrum of the discrete Laplacian on a Cayley graph of the group of intermediate growth constructed by me in 1980 (joint result with A.Dudko). Also, the case of the so called overgroup of intermediate growth will be discussed. The arguments will be based on a combination of results from algebra, representation theory, graph theory and classical harmonic analysis. This talk is intended to be easy for the audience, and everything will be quite elementary. The talk could also be considered as a preparation for a second, more advanced future talk of the speaker at GD seminar. |

Date: | October 18, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Robin Tucker-Drob, Texas A&M |

Title: | Cocycle Superrigidity of Bernoulli shifts and Compact actions |

Abstract: | I will discuss two results about cocycle superrigidity; the first, which is joint work with Adrian Ioana, is that Bernoulli shift of any nonamenable, inner amenable group, is cocycle superrigid. The second, which is joint with Damien Gaboriau and Adrian Ioana, is that, under a mild strong ergodicity assumption, any left-right translation action of a product group Γ×Λ on a profinite or connected compact group is virtually cocycle superrigid. The proofs of both results use the framework of deformation/rigidity. |

Date: | October 24, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Rostyslav Kravchenko, Northwestern University |

Title: | Characteristic random subgroups and their applications |

Abstract: | The invariant random subgroups (IRS's) were implicitly used by Stuck and Zimmer in 1994 in their study of lattices in simple Lie groups of higher rank. Around year 2010 there appeared several important papers that dealt with IRS's, among them papers by Vershik, Grigorchuk and Bowen. The name itself was coined by Abert, Glasner and Virag, who generalized a classical theorem of Kesten to the case of IRS's. Since then IRS's were actively studied, in particular Bowen has investigated the set of IRS's of free groups of finite rank, and Glasner studied it for linear groups. We define the notion of characteristic random subgroups (CRS's) which are a natural analog of IRS’s for the case of the group of all automorphisms. We determine CRS's for free abelian groups of infinite rank and for free groups of finite rank. Using our results on CRS's of free groups we show that for groups of geometrical nature (like hyperbolic groups, mapping class groups and outer automorphisms groups) there are infinitely many continuous ergodic IRS's. This is a joint work with L. Bowen and R. Grigorchuk. |

Date: | October 25, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Guoliang Yu, Texas A&M |

Title: | Dynamic dimension and K-theory |

Abstract: | I will introduce a notion of dynamic asymptotic dimension for group actions and discuss its application to K-theory. This is joint work with Erik Guentner and Rufus Willett. I will make the talk accessible to graduate students. |

Date: | November 1, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Volodymyr Nekrashevych, Texas A&M |

Title: | Etale groupoids, hyperbolic dynamics, and dimension |

Abstract: | We will discuss etale groupoids associated with hyperbolic dynamical systems, their structure, and applications to group theory and operator algebras. In particular, we will give a simple proof of finitness of their tower dimension (as defined by Kerr) and asymptotic dimension (as defined by Guentner, Willet, and Yu). |

Date: | November 8, 2017 |

Time: | 3:00pm |

Location: | BLOC 220 |

Speaker: | Ben Ben Liao, Texas A&M |

Title: | Noncommutative maximal inequalities for group actions |

Abstract: | Let $G$ be a finitely generated group, and $M$ a semi-finite von Neumann algebra on which $G$ acts. When the group $G$ has polynomial growth, we obtain strong type $(p,p),p>1,$ and weak type $(1,1)$ maximal inequalities for $G$ acting on $M$. This extends the results of Yeadon and Junge-Xu for the integer group. Based on joint work with Guixiang Hong and Simeng Wang. |