Activities This Week

Cornulier conjectured that two completely solvable Lie groups are quasiisometric if and only if they are isomorphic. This is a very difficult and very open problem. In this talk I will present some of the structure theory of solvable Lie groups and focus on the importance of sublinear distortions to this theory. I will review some of the important work of Cornulier and Tessera on Dehn functions of these groups. Finally, I will present recent results that contribute to their QI classification, which are interesting (also) because they are based on a (sublinear) weak form of QI.

Based on joint work with Gabriel Pallier.

נסתכל על חבורה קומפקטית $G$. דוגמה טובה היא חבורת כל ההעתקות האורתוגונליות במרחב:

\[G=SO(3)=\{U\in M_{3\times 3}(\mathbb{R})|UU^t=I\}\]

בהנתן שני איברים $a,b\in G$, נדון בשאלה: האם אפשר למצוא סדרה של ביטויים ב-$a,b$ שבהכרח מתכנסת ליחידה (בלי תלות בבחירה של $a,b$).

For a number field $K$ admitting an embedding into the field of real numbers $\mathbb{R}$, it is impossible to construct a functorial in $G$ group structure in the Galois cohomology pointed set $H^1(K,G)$ for all connected reductive $K$-groups $G$. However, over an arbitrary number field $K$, we define a diamond (or power) operation of raising to power $n$

We show that this operation has many functorial properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $x^{\Diamond n}$ coincides with the $n$-th power of $x$ in this group.

For a cohomology class $x \in H^1(K,G)$, we define the period $\operatorname{per}(x)$ to be the greatest common divisor of $n>0$ such that $x^{\Diamond n}=1$, and the index $\operatorname{ind}(x)$ to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $x$. These period and index generalize the period and index of a central simple algebra over $K$ (in the special case where $G$ is the projective linear group $\operatorname{PGL}_n$, the elements of $H^1(K, G)$ can be represented by central simple algebras). For an arbitrary reductive group $G$ defined over a local or global field $K$, we show that $\operatorname{per}(x)$ divides $\operatorname{ind}(x)$, that $\operatorname{per}(x)$ and $\operatorname{ind}(x)$ have the same prime factors, but the equality $\operatorname{per}(x) = \operatorname{ind}(x)$ may not hold.

The talk is based on joint work with Zinovy Reichstein. All necessary definitions will be given, including the definition of the Galois cohomology set $H^1(K,G)$.