Admissible Higson-Roe sequences for transformation groupoids
Moulay-Tahar Benameur, Victor Moulard
TL;DR
This work constructs a universal analytic six-term Higson–Roe sequence for transformation groupoids ${\mathcal G}=X\rtimes\Gamma$, valid for any admissible crossed-product completion and for metrizable finite-dimensional compact $\Gamma$-spaces $X$. It develops dual Roe algebras via a generalized Connes–Skandalis Hilbert module, proves independence of the $K$-theory of these algebras from the chosen representation, and establishes Paschke–Higson dualities that identify boundary maps with Baum–Connes assembly maps. By assembling these constructions across all proper cocompact $\Gamma$-spaces $Z\subset {\underline E}\Gamma$ and passing to inductive limits, the paper defines universal structure groups ${\mathcal S}_i(\Gamma,X)$ and yields a universal six-term sequence that specializes to the classical maximal and reduced Higson–Roe sequences and to the rectified BC framework when the crossed product is the BGW minimal exact Morita compatible version. The results provide a flexible framework to study rigidity phenomena for rho-invariants, connecting $K$-theory of dual Roe algebras, assembly maps, and structure groups in a way that adapts to the chosen crossed-product completion. These constructions pave the way for new rigidity results for group actions with rectified BC assembly maps and for potential extensions to broader transformation groupoids.
Abstract
Given a finitely generated discrete group Γ, we construct for any admissible crossed product completion and for any metrizable finite dimensional compact Γ-space X, a universal Higson-Roe six-term exact sequence for the transformation groupoid X\rtimes Γ. In particular, we generalize the maximal Higson- Roe sequence to such groupoids. In the case where the groupoid X\rtimes Γ satisfies the rectified Baum-Connes conjecture, this yields some rigidity consequences.