Jose Pedro Qintanilha - Geometric invariants for locally compact groups

Abstract

Talk

• Is there any connection we can make to MCGs?
• Can we link the invariants to different invariants (of MCGs?)

We are interested in $\Sigma$-invariants, BNSR invaraints, for Bieri, Neumann, Strebel & Benz

Finiteness Properties of groups

They generalise Finitely generated group and Finitely presented group. $G$ is of type $F_n$ if there is a $K(G,1)$ (K(G, 1) space) with finite $n$-skeleton.

A group is of type $F_n$ iff. there is a free $\Gamma$-complex it acts on with nice properties.

1. The zero skeleton is just $G$ (like a Cayley-Graph)
2. All $n$-cells have finitely many orbits under $G$ called $G$-cells.
3. It is $n-1$ connected

For $n=1$ we it is like a Cayley graph. For $n>1$ its similar but the $S$ does not need to be a generated set (it doesn’t need to be connected as a $1$-skeleton)

$G$ is of type $F_1$ iff. its finitely generated $G$ is of type $F_2$ iff. its finitely presented (finitely many $2$-cells)

He might be interested in Graphicayley :)

We will define geometric invariants, (i.e. BNSR inv. $\Sigma$-sets) as subsets of $Hom(G,\mathbb{R})$. (This remminds me of Clara Löh - L2-invariants of groups or Brita Nucinkis - Cohomological finiteness properties for totally disconnected locally compact groups)

Given a Character $\chi :G\to \mathbb{R}$. We define $G_{\chi }:=\{g\in G:\chi (g)\ge 0\}$. (Is this some kind of Left-orderable group?).

Definition

If $Y$ is a CW-complex with zero skeleton $G$ write $Y_{\chi }$ for the maximal subcomplex if $Y$ with zero skeleton $G_{\chi }$.

We can do an analogous construction for Quasimorphism. The ${\Sigma }^n$-set is this set of homology.

Theorem

$G$ is of type $F_n$ iff. $0\in {\Sigma }^n$. We can show that is ${\Sigma }^n$ is not empty, then it contains $0$.

And if $\chi \in {\Sigma }^n$ iff $\lambda \cdot \chi \in {\Sigma }^n$.

Theorem (BNSR)

${\Sigma }^n$ is the cone over an open subset of $Hom(G,\mathbb{R})\backslash \{0\}$.

Example

$\chi \in {\Sigma }^n⟺$ There is a finite subset $S\subseteq G$ s.t. $Cay(G,S{)}_{\chi }$ is connected. We show an example of ${\mathbb{Z}}^2$ and $F_2$. In one the character does lie in ${\Sigma }^1(F_2)$.

In fact: ${\Sigma }^1({\mathbb{Z}}^2)=Hom({\mathbb{Z}}^2,\mathbb{R}),{\Sigma }^1(F_2)=\{0\}$.

The following is the main theorem of sigma calculus! It allows us to propagate finiteness properties to subgroups.

Theorem (BNSR)

Suppose Short exact sequence $1\to N\to G\to Q\to 1$ with $Q$ abelian. If ${\Sigma }^n(G)$ contains all characters that vanish on $N$ (in particular $G$ is of type $F_n$). Then $N$ is of type $F_n$.

Compactness properties

Compactness properties are generalisations of finite properties for Hausdorff Topological groups.

Definition

Property $C_1$: The group is compactly generated. Property $C_2$: The group is compactly presented. (Relator length are bounded)

Theorem

For discrete groups $F_n⟺C_n$.(via Browns criterion.)

Theorem

If $H\subseteq G$ closed and cocompact (e.g. $H$ is a uniform Lattice (Lie group)) then $H$ and $G$ have the same compactness properties.

The core examples are “totally disconnected locally compact groups”, like Discrete group, profinite $S{L}_n({\mathbb{Q}}_p)$ Aut(locally finite graph)

The contribution of Jose was to define the sigma invariants ${\Sigma }_{top}^n\subseteq Ho{m}_{top}(G,\mathbb{R})$.

I just realised that the homomorphisms and the cohomology is related just as quasimorphisms and bounded cohomology are related.

Definition

Let $G$ be a locally compact Hausdorff group and $EG$ a free Simplicial set on $G$ (a souped up version of a simplicial complex).
The $k$-simplices of this set are tuples in $G$. Given a Character $\chi :G\to \mathbb{R}$, we consider a Filtration $(G_{\chi }\cdot EC{)}_{C\in \mathcal{C}}$ of $EG$. Now ${\Sigma }_{top}^n(G):=\{\chi :G\to \mathbb{R}|(G_{\chi }\cdot EC{)}_C\text{ is essentially (n−1)-connected}\}$

For discrete groups this agrees with ${\Sigma }^n(G)$. For other groups it still behaved similarly.