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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 1

### Equivariant $K$-theory of central extensions and twisted equivariant $K$-theory: $SL_{3}\mathbb{Z}$ and $St_{3}{\mathbb{Z}}$

Pages: 49 – 70

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n1.a4

#### Authors

#### Abstract

We compare twisted equivariant $K$-theory of $SL_{3}{\mathbb{Z}}$ with untwisted equivariant $K$-theory of a central extension $St_3{\mathbb{Z}}$. We compute all twisted equivariant $K$-theory groups of $SL_{3}{\mathbb{Z}}$, and compare them with previous work on the equivariant $K$-theory of $BSt_3{\mathbb{Z}}$ by Tezuka and Yagita.

Using a universal coefficient theorem by the authors, the computations explained here give the domain of Baum–Connes assembly maps landing on the topological $K$-theory of twisted group $C^*$-algebras related to $SL_{3}{\mathbb{Z}}$, for which a version of $KK$-theoretic duality studied by Echterhoff, Emerson, and Kim is verified.

#### Keywords

twisted equivariant $K$-theory, Bredon cohomology, Baum–Connes conjecture with coefficients, twisted group $C^*$-algebra, $KK$-theoretic duality

#### 2010 Mathematics Subject Classification

19K33, 19L47, 19L64