• Corpus ID: 249926905

# AH conjecture for Cantor minimal dihedral systems

```@inproceedings{Scarparo2022AHCF,
title={AH conjecture for Cantor minimal dihedral systems},
author={Eduardo Scarparo},
year={2022}
}```
. The AH Conjecture relates the low-dimensional homology groups of a groupoid with the abelianization of its topological full group. We show that transformation groupoids of minimal actions of the inﬁnite dihedral group on the Cantor set satisfy this conjecture. The proof uses Kakutani-Rokhlin partitions adapted to such systems.
1 Citations

### Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

. Inspired by work of Szymik and Wahl on the homology of Higman-Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and

## References

SHOWING 1-10 OF 19 REFERENCES

### On continuous orbit equivalence rigidity for virtually cyclic group actions

We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they

### A counterexample to the HK-conjecture that is principal

• R. Deeley
• Mathematics
Ergodic Theory and Dynamical Systems
• 2022
Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a

### Katsura–Exel–Pardo groupoids and the AH conjecture

• Mathematics
Journal of the London Mathematical Society
• 2021
It is proven that Matui's AH conjecture is true for Katsura–Exel–Pardo groupoids GA,B associated to integral matrices A and B . This conjecture relates the topological full group of an ample groupoid

### Almost finiteness and homology of certain non-free actions

• Mathematics
• 2020
We show that Cantor minimal \$\mathbb{Z}\rtimes\mathbb{Z}_2\$-systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal

### Matui's AH conjecture for Graph Groupoids

• Mathematics
• 2020
We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a conjecture which relates the topological full group of an ample groupoid with the homology of the groupoid.

### Full groups of Cantor minimal systems

• Mathematics
• 1999
We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence

### Reversing and extended symmetries of shift spaces

• Mathematics
• 2016
The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity

### Homology of odometers

We compute the homology groups of transformation groupoids associated with odometers and show that certain \$(\mathbb{Z}\rtimes \mathbb{Z}_{2})\$-odometers give rise to counterexamples to the HK

### CROSSED-PRODUCTS OF TOTALLY DISCONNECTED SPACES BY Z(2)*Z(2)

• Mathematics
• 2018
Let Q be a totally disconnected compact metrizable space, and let (t be a minimal homeomorphism of Q. Let 0" be a homeomorphism of order 2 on Q such that (to" = O"(t-l, and assume that 0" or (to" has

### ON ALGEBRAIC PROPERTIES OF TOPOLOGICAL FULL GROUPS

In the paper we discuss the algebraic structure of topological full group [[T ]] of a Cantor minimal system (X,T ). We show that the topological full group [[T ]] has the structure similar to a union