# Groups with Spanier-Whitehead duality

## TL;DR

This paper explores Spanier-Whitehead K-duality for discrete groups, linking it to the Baum-Connes conjecture, and proves duality for various classes of groups including Bieberbach's space groups and lattices in Lorentz groups.

## Contribution

It introduces a new perspective on K-duality for groups, comparing methods and establishing duality results for broad classes of groups.

## Key findings

- Proves Spanier-Whitehead duality for Bieberbach's space groups.
- Establishes duality for groups acting on trees.
- Confirms duality for lattices in Lorentz groups.

## Abstract

Building on work by Kasparov, we study the notion of Spanier-Whitehead K-duality for a discrete group. It is defined as duality in the KK-category between two C*-algebras which are naturally attached to the group, namely the reduced group C*-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.