Abstract:
Around 2015, Hiroki Matui introduced two conjectures concerning minimal étale groupoids over Cantor spaces. Such groupoids serve as models for a great many unital simple C*-algebras. The HK-conjecture relates the homology groups of the groupoid with the K-groups of its reduced groupoid C*-algebra. Whereas the AH-conjecture relates the homology groups of the groupoid with (the abelianization of) its topological full group. I will give an introduction to these two conjectures, report on their current status, and, if time permits, say a few words of my own research on the AH-conjecture.