Etale equivalence relations with certain prescribed torsion in their homology
Michael Francesco Ala, Hung-Chang Liao, Aaron Tikuisis
TL;DR
This work constructs minimal étale equivalence relations whose zeroth homology decomposes as a direct sum $H_0(\mathcal{R}) \cong D \oplus E$ with $D$ a simple dimension group and $E \subset \mathbb{Q}/\mathbb{Z}$. The method starts from a Bratteli diagram realizing $D$, then performs a vertex-splitting procedure and augments tail-equivalence by a carefully designed partial homeomorphism to encode $E$, yielding explicit realizations such as $\mathbb{Z}[\tfrac{1}{2}] \oplus \mathbb{Z}/2\mathbb{Z}$ and a Fibonacci-type dimension group with a $\mathbb{Z}/2\mathbb{Z}$ summand. The paper develops a general procedure (Bratteli splitting, path-space recipes, and compatibility criteria) to realize arbitrary $D \oplus E$ in $H_0(\mathcal{R})$, and proves that the resulting relations can be almost finite and, under symmetry conditions, possess approximately inner flip. These results broaden the spectrum of invariants achievable by minimal étale equivalence relations and connect dimension-group realizations with torsion phenomena observed in tiling-like dynamics.
Abstract
Given a non-cyclic simple dimension group D and a subgroup E of Q/Z, we produce a minimal étale equivalence relation R such that H_0(\R) is isomorphic to D \oplus E, where H_0(R) denotes the zeroth homology group of R. The equivalence relation R arises by combining tail-equivalence on a Bratteli diagram with a partial homeomorphism.