Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture

Abstract: In this paper, we study 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 $\alpha$ by a countable discrete group $\Gamma$, if $X$ admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space and $\Gamma$ is torsion-free, then the equivariant coarse strong Novikov conjecture holds rationally for $(X, \Gamma, \alpha)$. In the second part, we drop the torsion-free assumption for $\Gamma$ in the first part. We introduce a Miscenko-Kasparov assembly map for a proper $\Gamma$-space $X$ with equivariant bounded geometry, which leads to a new Novikov-type conjecture that we call the analytic equivariant coarse Novikov conjecture. We show that for a proper $\Gamma$-space $X$ with equivariant bounded geometry, the rational analytic equivariant coarse Novikov conjecture holds for $(X,\Gamma,\alpha)$ if $X$ admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space, i.e., the Miscenko-Kasparov assembly map is a rational injection.