Brita Nucinkis - Cohomological finiteness properties for totally disconnected locally compact groups

Abstract

The aim of this course is to introduce cohomological finiteness conditions both from an algebraic and a geometric viewpoint. I will begin by introducing the main concepts, such as cohomological dimension and cohomological finiteness lengths for discrete groups, and give an overview of some interesting examples. I will also introduce Brown’s criterion, which is a powerful tool to determine the cohomological finiteness length. Many of these concepts can be extended to study finiteness conditions for totally disconnected locally compact groups. I will give an overview of current results and open questions.

Talk

Minitalk I

1. Introduction

Main questions:

• What is a finiteness property?
• What about topological groups?

Definition

A finiteness property is a property $P$ that is satisfied by a group admitting a finite $K(G,1)$ (?)

So what are these finiteness conditions?

• geometric dimension: The smallest dimension of a model for EG (the Classifying space?)
• cocompact EG
• finite generation
• finite presentation
• finite groups

These are things which describe “finiteness” in some sense.

Definition

A group is of type $F_n$ if there is a $K(G,1)$ with finitely many $k$-cells with $k\le n$ We say, it is of type infinity $F_{\infty }$ if it is $F_n$ for all $n$.

See K(G, 1) space.

Example

• Every group is $F_0$.
• $G$ is $F_1$ if it is Endlich Erzeugte Gruppe
• $G$ is $F_2$ if it is Endlich Präsentierbare Gruppe

I finally get it.

2. Algebraic counterpart: Group cohomology

We take $k$ to be a Commutative Ring and we study a long exact sequence, ending with $...P_0\to \to k\to 0$ where all spaces $P_i$ consist of projective modules.

We continue by defining the property $F{P}_n(k)$ and $F{P}_{\infty }(k)$. They are defined by a Projective resolution.

Theorem

A group is finitely generated iff. $G$ is of type $FP$. If it is finitely presented then $G$ is of type $F{P}_2$. The converse does not hold.

Theorem (Stalling-Swan 60, Eilenberg-Ganea 57)

There is cohomological dimension and geometrical dimension. They are defined differently and until now it is not known whether they are always equal but in most cases they are.

Question: Are there groups of type FL but not FP?

Theorem (Wall 66)

Let $G$ be f.p. and FL, then $G$ admits a cocompact model EG.

From this follows: $G$ is f.p. and $F{P}_n$ iff. $G$ is of type $F_n$.

Question: Find families of groups of type $F{P}_n$ but not of $F{P}_{n+1}$ for all $n$. This is actually a really nice thing to study.

Partial examples to this questions are given by:

Example

• Stallings 63: $F{P}_2$ but not $F{P}_3$
• Bieri 76: $F{P}_n$ but not $F{P}_{n+1}$ isn’t this just the question?
• Bestina-Brody 97: Proposition to that question
• Abels-Brown 92: Found a family of S-arithmetic groups
• Skipper-Weitel-Zarmsky 98: Family of Einfache Gruppe.

3. HNN-extensjions & Mayer-Vietoris-sequences

We define an HNN-Erweiterung. We give a Mayer-Vietoris sequence-construction made out of HNN-extensions.

Book to learn this: A book published at queensmary college.

They showed that if $G$ is of type $F{P}_n$ then the Homologiegruppe $H^k(G,\mathbb{Z})$ is f.g.