The coarse Baum-Connes conjecture for certain relative expanders

Abstract

Let (1 → Nn → Gn → Qn → 1) n∈N be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence (Nn)n∈N with the induced metric from the word metrics of (Gn) n∈N has property A, and the sequence (Qn) n∈N with the quotient metrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence (Gn) n∈N , which may not admit a coarse embedding into Hilbert space. It follows that the coarse Baum-Connes conjecture holds for the relative expanders and group extensions exhibited by G. Arzhantseva and R. Tessera, and special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This in particular solves an open problem raised by G. Arzhantseva and R. Tessera [3]. Contents

K-theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations

“…Nevertheless, the proof of the recent result in [12] by J. Deng, Q. Wang and G. Yu for sequences of group extensions with A-by-CE structure can be adapted to metric spaces with A-by-CE coarse fibration. More precisely, we have the following result.…”

“…Theorem 1.3 (J. Deng, Q. Wang and G. Yu [12]). Let X be a discrete metric space with bounded geometry.…”

“…Very recently, J. Deng, Q. Wang and G. Yu [12] proved that, for a sequence of group extensions (1 → N n → G n → Q n → 1) n∈N of finitely generated groups with uniformly finite generating subsets, if the coarse disjoint union of (N n ) n∈N has Yu's property A and the coarse disjoint union of (Q n ) n∈N admits a coarse embedding into Hilbert space, then the coarse Baum-Connes conjecture holds for the coarse disjoint union of (G n ) n∈N . Such a sequence of group extensions…”

“…The proof of Theorem 1.2 relies on the work of J. Deng, Q. Wang and G. Yu [12]. We will mainly focus on the proof for the (maximal and reduced) uniform Roe algebras, i.e.…”

The coarse Novikov conjecture for extensions of coarsely embeddable groups

“…In [10], J. Deng, Q. Wang and G. Yu raise an open problem: Does "CE-by-CE" imply the coarse Baum-Connes conjecture?…”