## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 63, Number 2 (2019), 317-334.

### K-theory and K-homology of finite wreath products with free groups

#### Abstract

This article investigates an explicit description of the Baum–Connes assembly map of the wreath product $\Gamma =F\wr {\mathbb{F}}_{n}={\oplus}_{{\mathbb{F}}_{n}}F\u22ca{\mathbb{F}}_{n}$, where $F$ is a finite and ${\mathbb{F}}_{n}$ is the free group on $n$ generators. In order to do so, we take Davis–Lück’s approach to the topological side which allows computations by means of spectral sequences. Besides describing explicitly the K-groups and their generators, we present a concrete 2-dimensional model for the classifying space $\underline{\mathrm{E}}\Gamma $. As a result of our computations, we obtain that ${\mathrm{K}}_{0}\left({\mathrm{C}}_{\mathrm{r}}^{\ast}\right(\Gamma \left)\right)$ is the free abelian group of countable rank with a basis consisting of projections in ${\mathrm{C}}_{\mathrm{r}}^{\ast}\left({\oplus}_{{\mathbb{F}}_{n}}F\right)$, and ${\mathrm{K}}_{1}\left({\mathrm{C}}_{\mathrm{r}}^{\ast}\right(\Gamma \left)\right)$ is the free abelian group of rank $n$ with a basis represented by the unitaries coming from the free group.

#### Article information

**Source**

Illinois J. Math., Volume 63, Number 2 (2019), 317-334.

**Dates**

Received: 8 November 2018

Revised: 12 April 2019

First available in Project Euclid: 1 August 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1564646437

**Digital Object Identifier**

doi:10.1215/00192082-7768735

**Mathematical Reviews number (MathSciNet)**

MR3987499

**Zentralblatt MATH identifier**

07088309

**Subjects**

Primary: 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]

Secondary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20]

#### Citation

Pooya, Sanaz. K-theory and K-homology of finite wreath products with free groups. Illinois J. Math. 63 (2019), no. 2, 317--334. doi:10.1215/00192082-7768735. https://projecteuclid.org/euclid.ijm/1564646437