פעילויות השבוע

אני אנסה להציג, בצורה מאוד לא פורמלית, כמה צדדים של תורת המודלים, תחום בלוגיקה מתמטית המתמקד במבנה של קבוצות ופונקציות המוגדרות על-ידי נוסחאות בלוגיקה מסדר ראשון. לתחום שימושים מפתיעים במגוון ענפים במתמטיקה, ואני אציג אותו דרך משפחה של שימושים שקשורים בפתרונות של משוואות פולינומיות. ההרצאה לא תניח ידע קודם בלוגיקה או בפתרון משוואות פולינומיות (ולכן הדגש יהיה על אינטואיציה ולא על טענות מדויקות)

This is joint work with Ilan Hirshberg.

Recall that an action of a group G on a set X is free if every nontrivial group element acts with no fixed points, such as the action of G on itself by translation.

Now assume G is compact, and consider an action of G on a unital C*-algebra. Finite Rokhlin dimension (with commuting towers, the only version we consider here) is one of several possible noncommutative versions of freeness. To put it in context, the action of G on G \times X by translation in the first variable has Rokhlin dimension zero, while the action x \mapsto - x of the two element group Z_2 on the sphere S^d has Rokhlin dimension d.

What are the possible values of Rokhlin dimension for actions on simple C*-algebras? Rokhlin dimension zero is just the (much older) Rokhlin property, and many examples are known. Several examples are known with Rokhlin dimension exactly one, and one with Rokhlin dimension exactly two, but until now no examples were known to have Rokhlin dimension finite but greater than two, even without knowing the exact value.

We construct actions of finite groups and the circle S^1 on simple C-algebras, in some cases even simple AF algebras, for which we can prove that the Rokhlin dimension is large but finite. One of our most precise results is that for every positive even integer d, there is an action of Z_2 on a simple unital AF algebra whose Rokhlin dimension is exactly d. However, much remains open about the possible values of Rokhlin dimension for actions of compact groups on simple C-algebras.

We generalize the combinatorial approaches of Rapaport and Higgins–Lyndon to the Whitehead algorithm. We show that for every automorphism φ of a free group F and every word u∈F there exists a finite multiset of words Su,φ satisfying the following property: For every cyclic word w, the number of times u appears as a subword of φ(w) depends only on the appearances of words in Su,φ as subwords of w. We use this fact to construct a faithful representation of Out(Fn) on an inverse limit of Z-modules, so that each automorphism is represented by sequence of finite rectangular matrices, which can be seen as successively better approximations of the automorphism.

Classical Szpilrajn theorem states that any partial order could be extended to a linear order. An invariant random order (IRO) on a countable group is an invariant under the shift-action probability measure on the space of all partial orders on the group. It is natural to ask whether the invariant analog of Szpilrajn theorem, the invariant random order extension property, holds for IRO‘s. This property is easy to demonstrate for amenable groups. Recently, Glasner, Lin a Meyerovitch gave a first example where this property fails. Based on their construction, I will show that the IRO extension property fails for all non-amenable groups.