The equivariant coarse Baum-Connes conjecture for metric spaces with proper group actions

TL;DR

This paper proves the equivariant coarse Baum-Connes conjecture for certain metric spaces with group actions, under conditions like amenability, coarse embeddability into Hilbert space, and uniform orbit equivalence.

Contribution

It establishes the conjecture for spaces with proper group actions satisfying specific geometric and algebraic conditions, extending previous results.

Findings

The conjecture holds when the quotient space admits a coarse embedding into Hilbert space.

The group $ ext{amenable}$ and orbit conditions are sufficient for the conjecture to hold.

A $K$-theoretic amenability result for the $ ext{Gamma}$-space $X$ is proven.

Abstract

The equivariant coarse Baum-Connes conjecture interpolates between the Baum-Connes conjecture for a discrete group and the coarse Baum-Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group $\Gamma$ acts properly and isometrically on a discrete metric space $X$ with bounded geometry, not necessarily cocompact. We show that if the quotient space $X/\Gamma$ admits a coarse embedding into Hilbert space and $\Gamma$ is amenable, and that the $\Gamma$-orbits in $X$ are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum-Connes conjecture holds for $(X, \Gamma)$. Along the way, we prove a $K$-theoretic amenability statement for the $\Gamma$-space $X$ under the same assumptions as above, namely, the canonical quotient map from the maximal equivariant Roe algebra of $X$ to the reduced equivariant Roe algebra of $X$ induces an isomorphism on $K$-theory.