Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture

Abstract

The equivariant coarse Novikov conjectures stand among a handful profound $K$-theoretic conjectures in noncommutative geometry. Motivated by the quest to verify Novikov-type conjectures for groups of diffeomorphisms, we study in this paper the equivariant coarse Novikov conjectures for spaces that equivariantly and coarsely embed into admissible Hilbert-Hadamard spaces, which are a type of infinite-dimensional nonpositively curved spaces. The paper is split into two parts. We prove in the first part that for any metric space $X$ with bounded geometry and with a proper isometric action $α$ by a countable discrete group $Γ$, if $X$ admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space and $Γ$ is torsion-free, then the equivariant coarse strong Novikov conjecture holds rationally for $(X, Γ, α)$. In the second part, we extend the result in the first part by dropping the torsion-free assumption on $Γ$. To this end, we introduce, for a proper $Γ$-space $X$ with equivariant bounded geometry, a new Novikov-type conjecture that we call the rational analytic equivariant coarse Novikov conjecture, which generalizes the rational analytic Novikov conjecture and asserts the rational injectivity of a certain assembly map associated with a coarse analog of the classifying space $EΓ$. We show that for a proper $Γ$-space $X$ with equivariant bounded geometry, if $X$ admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space, then the rational analytic equivariant coarse Novikov conjecture holds for $(X,Γ,α)$, i.e., the assembly map is a rational injection.