The K-theory of maximal and reduced Roe algebras for Hecke pairs with equivariant coarse embeddings

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.

💡 Research Summary

The paper studies the K‑theoretic relationship between maximal and reduced equivariant Roe algebras associated with Hecke pairs ((\Gamma ,\Lambda )). A Hecke pair consists of a countable discrete group (\Gamma) and an almost normal subgroup (\Lambda); the almost normal condition is shown to be equivalent to the finiteness of left‑action orbits of (\Lambda) on the coset space (\Gamma /\Lambda). The authors first establish that when the quotient space (X=\Gamma /\Lambda) admits a (\Gamma)-equivariant coarse embedding into a Hilbert space, the stabilizer of any point in the Hilbert space is commensurable with (\Lambda). This geometric “cut‑and‑paste’’ argument relies on the bounded geometry of (X) and the control functions (\rho_{\pm}) coming from the coarse embedding.

With this commensurability in hand, the paper defines both reduced and maximal equivariant Roe algebras for a proper, cocompact (\Gamma)-action on a proper metric space, and introduces a twisted version that generalizes crossed products. The central technical achievement is the adaptation of Yu’s Dirac‑dual‑Dirac method to the Hecke‑pair setting. Assuming (\Lambda) is a‑T‑menable (i.e., has the Haagerup property), the authors construct Dirac and dual‑Dirac elements in Kasparov’s (KK)-theory for (\Lambda) and then lift them to (\Gamma) via Kasparov products. Because any subgroup commensurable with (\Lambda) inherits a‑T‑menability, every such subgroup is K‑amenable, which yields the crucial equality \