C5

Let $A$ and $B$ be boundaries of CAT(0) spaces. A function $f:A \to B$ is called a {\em boundary isomorphism} if $f$ is a homeomorphism in the visual topology and

$f$ is an isometry in the Tits metric. A compact metrizable space $Y$ is said to be {\em Tits rigid}, if for any two CAT(0) group boundaries $Z_1$ and $Z_2$ homeomorphic to $Y$, $Z_1$ is boundary isomorphic to $Z_2$.

We prove that the join of two Cantor sets and its suspension are Tits rigid.

A virtual endomorphism of a group $G$ is a homomorphism $f : H \rightarrow G$ where $H$

is a subgroup of $G$ of fi…nite index $m$: A recursive construction using $f$ produces a

so called state-closed (or, self-similar in dynamical terms) representation of $G$ on

a 1-rooted regular $m$-ary tree. The kernel of this representation is the $f$-core $(H)$;

i.e., the maximal subgroup $K$ of $H$ which is both normal in G and is f-invariant.

Examples of state-closed groups are the Grigorchuk 2-group and the Gupta-

Sidki $p$-groups in their natural representations on rooted trees. The affine group

$Z^n \rtimes GL(n;Z)$ as well as the free group $F_3$ in three generators admit state-closed

representations. Yet another example is the free nilpotent group $G = F (c; d)$ of

class c, freely generated by $x_i (1\leq i \leq d)$: let $H = \langle x_i^n | \

(1 \leq i \leq d) \rangle$ where $n$ is a

fi…xed integer greater than 1 and $f$ the extension of the map $x^n_i

\rightarrow x_i$ $(1 \leq i \leq d)$.

We will discuss state-closed representations of general abelian groups and of

…nitely generated torsion-free nilpotent groups.

Let G be a finite group, p a prime and S a Sylow p-subgroup. The group G

is called p-nilpotent if S has a normal complement N in G, that is, G is

the semidirect product between S and N. The notion of p-nilpotency plays

an important role in finite group theory. For instance, Thompson's

criterion for p-nilpotency leads to the important structural result that

finite groups with fixed-point-free automorphisms are nilpotent.

By a classical result of Tate one can detect p-nilpotency using mod p

cohomology in dimension 1: the group G is p-nilpotent if and only if the

restriction map in cohomology from G to S is an isomorphism in dimension

1. In this talk we will discuss cohomological criteria for p-nilpotency by

Tate, and Atiyah/Quillen (using high-dimensional cohomology) from the

1960s and 1970s. Finally, we will discuss how one can extend Tate's

result to study p-solvable and more general finite groups.

A finitely generated group has the Haagerup property if it admits a proper isometric action on a Hilbert space. It was a long open question whether Haagerup property is a quasi-isometry invariant. The negative answer was recently given by Mathieu Carette, who constructed two quasi-isometric generalized Baumslag-Solitar groups, one with the Haagerup property, the other not. Elaborating on these examples, we proved (jointly with S. Arnt and T. Pillon) that the equivariant Hilbert compression is not a quasi-isometry invariant. The talk will be devoted to describing Carette's examples.

(Joint work with Jochen Koenigsmann) Admitting a p-henselian
valuation is a weaker assumption on a field than admitting a henselian
valuation. Unlike henselianity, p-henselianity is an elementary property
in the language of rings. We are interested in the question when a field
admits a non-trivial 0-definable p-henselian valuation (in the language
of rings). They often then give rise to 0-definable henselian
valuations. In this talk, we will give a classification of elementary
classes of fields in which the canonical p-henselian valuation is
uniformly 0-definable. This leads to the new phenomenon of p-adically
(pre-)Euclidean fields.

A universal D-module of dimension n is a rule assigning to every family of smooth $n$-dimensional varieties a family of D-modules, in a compatible way. This seems like a huge amount of data, but it turns out to be entirely determined by its value over a single formal disc. We begin by recalling (or perhaps introducing) the notion of a D-module, and proceed to define the category $M_n$ of universal D-modules. Following Beilinson and Drinfeld we define the Gelfand-Kazhdan structure over a smooth variety (or family of varieties) of dimension $n$, and use it to build examples of universal D-modules and to exhibit a correspondence between $M_n$ and the category of modules over the group-scheme of continuous automorphisms of formal power series in $n$ variables

In an o-minimal expansion of the real field, while few holomorphic functions are globally definable, many may be locally definable. Wilkie conjectured that a few basic operations suffice to obtain all of them from the basic functions in the language, and proved the conjecture at generic points. However, it is false in general. Using Ax's theorem, I will explain one counterexample. However, this is not the end of the story.
This is joint work with Jones and Servi.