Rational equivariant K-homology of low dimensional groups

TL;DR

This paper develops an algorithm to compute the rationalized equivariant K-homology of groups with a 3-manifold classifying space, linking it to the K-theory of associated C*-algebras under certain geometric conditions.

Contribution

It introduces a new algorithm for calculating rationalized equivariant K-homology for low-dimensional groups with geometric structures, connecting it to C*-algebra K-theory.

Findings

Algorithm successfully computes rationalized equivariant K-homology for specific groups.

Under geometrizability, K-homology matches K-theory of the reduced C*-algebra.

Illustrative examples demonstrate the algorithm's practical application.

Abstract

We consider groups G which have a cocompact, 3-manifold model for the classifying space \underline{E}G. We provide an algorithm for computing the rationalized equivariant K-homology of \underline{E}G. Under the additional hypothesis that the quotient 3-orbifold \underline{E}G/G is geometrizable, the rationalized K-homology groups coincide with the rationalized K-theory of the reduced C*-algebra of G. We illustrate our algorithm on some concrete examples.