## The Lille workshop on Coarse Geometry of Groups

Here start notes taken during the workshop “Coarse Geometry of Groups”, Lille, june 18-22, 2012. Organizers: C. Drutu (Lille + Oxford), R. Tessera (Lyon).

Talks I missed on monday, june 18th

I missed the first day, 5 talks. I tried to reconstruct 3 of them from conversations with speakers. This must be full of mistakes, all my own.

Bader defines the Weyl group ${W}$ of a group ${\Gamma}$ as follows. Consider a Poisson boundary ${B}$ of ${\Gamma}$ (in Furstenberg’s sense). Assume that ${\Gamma}$ is ergodic on ${B \times B}$. Then ${W}$ is the centralizer of ${\Gamma}$ in the group of measure preserving transformations of ${B \times B}$.

Motivating example: ${\Gamma}$ a lattice in a semi-simple Lie group ${G}$. Then ${W}$ turns out to be the Weyl group of ${G}$, a finite group.

Poisson boundaries form a poset, with ${B\leq B'}$ if there is a ${\Gamma}$-equivariant map of ${B'}$ onto ${B}$. In the motivating example, this poset corresponds to the poset of parabolic subgroups of ${G}$. All these constructions are natural under group homomorphisms ${\Gamma \rightarrow \Gamma'}$.

Corollary 1 (Bader-Furman) Lattices in products cannot map to lower rank semi-simple groups.

2. Rufus Willett

Willett studies the coarse Baum-Connes conjecture. This relates K-theory of a space and K-theory of its Roe algebra (algebra of operators whose kernels vanish outside a bounded neighborhood of the diagonal). A notion of metric (T) property turns out to be an obstruction for surjectivity of the assembly map. To overcome it, one must modify the norm on Roe’s algebra, and introduce the maximum norm with respect to all unitary representations of the Roe algebra.

Metric (T) property states that the spectrum of a coarse Laplacian (e.g. simplicial Laplacian of Rips’ complex) is bounded below for every unitary representation of the Roe algebra.

Example 1 Let ${\Gamma}$ be a residually finite group, let ${X}$ be the disjoint union of ${\Gamma/\Gamma_n}$ where ${\Gamma_n}$ has finite index in ${\Gamma}$ and the intersection of all ${\Gamma_n}$ is trivial. Then

$\displaystyle \begin{array}{rcl} X \textrm{ has metric property (T) } \Leftrightarrow \Gamma \textrm{ has property (T)}. \end{array}$

Proposition 2 K-theory does change after change to the max norm if X admits a coarse embedding into a Hilbert space.

Guoliang Yu introduced Property A for metric spaces as a tool to construct coarse embeddings into Hilbert spaces. Conversely, it is hard to negate. Here is a consequence of Property A which is easier to negate.

Definition 3 Let ${X}$ be a bounded geometry metric space. ${X}$ is said to be uniformly locally amenable if for all ${R > 0}$ and ${\epsilon > 0}$, there exists ${S > 0}$ such that for any finite subset ${F}$ of ${X}$, there exists a subset ${E}$ if ${X}$ such that diam${(E)\leq S}$ and

$\displaystyle \begin{array}{rcl} | (E+r\setminus E) \cap F |\leq\epsilon | E \cap F |. \end{array}$

This is a coarse invariant.

3. Piotr Nowak

Nowak has a variant of Garland’s formula, used for proving fixed point properties of groups acting on metric spaces. His formula is less general than the one used previously, in that it applies only to Banach spaces (in fact, reflexive Banach spaces). However, it involves linear Poincaré inequalities, i.e.

$\displaystyle \begin{array}{rcl} || f - c || \leq C\, E(f). \end{array}$

where ${c}$ is the (ordinary, linear) average of ${f}$. Functional analytic techniques can be used to estimate the constant ${C}$. For instance, interpolation (between ${L^2}$ and ${L^\infty}$) gives an estimate for target space ${L^p}$.

Theorem 4 For groups for which property (T) can be proved using Garland’s formula, a fixed point theorem holds on ${L^p}$ for ${p}$ in an explicit interval containing 2.

The argument applies as well to non isometric but uniformly bounded (with norm close enough to 1) affine actions. The theorem implies vanishing of ${L^p}$ cohomology. This shows that no such theorem can hold for all ${p}$.

Example 2 – Groups acting cocompactly on ${\tilde{A}_2}$-buildings. – Random groups in the density model. Using Bourdon-Pajot, one gets a lower bound on their conformal dimensions.

4. Announcements

– September 24-29: Inaugural conference of CEMPI, European Center for Maths, Physics, Interactions.

Operator algebras: 24-26.

Graphs and groups: 27-29.

– September 2-9: Dynamics of homogeneous flows, Technion.

– September 25-29: CRM Summer school on ${Out(F_n)}$, Barcelona.