Groups and Dynamics Workshop

Texas A&M University

May 6, 2014, Blocker 624

**Organizers:** Rostislav Grigorchuk and Dmytro Savchuk

**Schedule:**

02:00pm-2:50pm | N. Romanovsky (Novosibirsk, Russia) |
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Logical Aspects of the Theory of Rigid Solvable Groups | |

Abstract: pdf | |

3:00pm-3:50pm | S. Rae (Texas A&M University) |

Finiteness Property of Wreath Products | |

Abstract: Let X be a G-set for a group G. Let A be another group. The permutational wreath product A \wr G is defined via the action of G on the direct sum of A (indexed by X) by permutation of factors. Yves de Cornulier characterized precisely when permutational wreath products are finitely presented. We consider a family of abelian-by-Houghton's groups to construct examples of wreath products with various finiteness properties. If time permits we also discuss related questions. | |

3:50pm-4:10pm | Coffee Break |

4:10pm-5:00pm | D. Savchuk (University of South Florida) |

Submanifold Projection for Out(F)_{n} | |

Abstract: One of the most useful tools for studying the geometry of the mapping class group has been the subsurface projections of Masur and Minsky. We propose an analogue for the study of the geometry of Out(Fn) called submanifold projection. We use the doubled handlebody $M_n=#_n S^2×S^1$ as a geometric model of $F_n$, and consider essential embedded 2-spheres in $M_n$, isotopy classes of which can be identified with free splittings of the free group. We interpret submanifold projection in the context of the sphere complex (also known as the splitting complex). We prove that submanifold projection satisfies a number of desirable properties, including a Behrstock inequality and a Bounded Geodesic Image theorem. This is a joint work with Lucas Sabalka. | |

5:10pm-6:00pm | K. Medynets (US Naval Academy) |

Applications of Topological Dynamics to Representation Theory of Transformation Groups | |

Abstract: Studying the representation theory of the infinite symmetric group S(N), Vershik noticed that "almost every" character $f$ of $S(N)$ came from an ergodic action of S(N) on a measure space $(X,\mu)$ by the formula $f(g) = \mu(FixedPoints(g))$. If this observation holds for a group in question, then using topological dynamics techniques one can potentially describe all characters.
Note that in view of GNS construction, classification of characters is equivalent to the classification of II_1 (in the sense of Murray-Neumann) representations of the group in question.
We will reexamine some older results on the classification of characters for such groups as Special Linear Group of Infinite Matrices over a finite field and Groups of Rational Permutation of the unit interval. We will then classify characters of locally finite groups determined by Bratteli diagrams and the Higman-Thompson groups. In the case of Higman-Thompson groups, we will show that these groups have no non-trivial characters. The absence of non-trivial characters have some implications in the theory of random subgroups. This talk is based on joint works with Artem Dudko. |