The K-theory of maximal and reduced Roe algebras for Hecke pairs with equivariant coarse embeddings
Liang Guo, Hang Wang, Xiufeng Yao
TL;DR
This work proves that for a Hecke pair $(\Gamma,\Lambda)$ with quotient $X=\Gamma/\Lambda$, if $X$ admits a $\Gamma$-equivariant coarse embedding into a Hilbert space and $\Lambda$ is a‑T‑menable, then $\Gamma$ is quasi $K$-amenable, yielding isomorphisms $K_*(B\rtimes\Gamma) \cong K_*(B\rtimes_r\Gamma)$ for all $\Gamma$-$C^*$-algebras $B$. The authors implement the Dirac–dual–Dirac framework in this Hecke-pair context, introducing Bott and Dirac asymptotic morphisms and a cutting‑and‑paste mechanism to reduce global questions to stabilizers commensurable with $\Lambda$. These techniques enable Baum–Connes with coefficients, the strong Novikov conjecture, and coarse Baum–Connes results for $\Gamma$ under the stated coarse-embedding hypotheses, with corollaries when both $\Lambda$ and $X$ coarsely embed. The results extend the stability of $K$-amenability under extensions to the Hecke-pair setting and provide a geometric route to several conjectures in noncommutative geometry for groups built from Hecke pairs. The paper thus broadens the applicability of the Dirac–dual–Dirac method in coarse geometry and operator algebras.
Abstract
In this paper, we generalize the Dirac-dual-Dirac method to Hecke pairs with equivariant coarse embeddings and establish the K-theoretic isomorphisms between the maximal and reduced equivariant Roe algebras. We also extend these results to prove the Baum--Connes conjecture in this context.