## Illinois Journal of Mathematics

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

Sanaz Pooya

#### Abstract

This article investigates an explicit description of the Baum–Connes assembly map of the wreath product $\Gamma=F\wr\mathbb{F}_{n}=\bigoplus_{\mathbb{F}_{n}}F\rtimes\mathbb{F}_{n}$, 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 $\mathrm{K}_{0}(\mathrm{C}^{*}_{\mathrm{r}}(\Gamma))$ is the free abelian group of countable rank with a basis consisting of projections in $\mathrm{C}^{*}_{\mathrm{r}}(\bigoplus_{\mathbb{F}_{n}}F)$, and $\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathrm{r}}(\Gamma))$ 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
Revised: 12 April 2019
First available in Project Euclid: 1 August 2019

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

Digital Object Identifier
doi:10.1215/00192082-7768735

Mathematical Reviews number (MathSciNet)
MR3987499

Zentralblatt MATH identifier
07088309

#### 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

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