[Paper Review] The coarse Baum-Connes conjecture for certain extensions and relative expanders

Jintao Deng, Qin Wang+1 more|arXiv (Cornell University)|Feb 21, 2021

Advanced Operator Algebra Research40 references4 citations

TL;DR

This paper establishes the coarse Baum-Connes conjecture for certain group extensions and relative expanders by leveraging property A for normal subgroups and coarse Hilbert space embeddability of quotient groups. It proves the conjecture holds even when the full group does not coarsely embed into Hilbert space, resolving an open problem regarding relative expanders and special box spaces of free groups.

ABSTRACT

Let $\left( 1 o N_n o G_n o Q_n o 1 ight)_{n\in \mathbb{N}}$ be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence $\left( N_n ight)_{n\in \mathbb{N}} $ with the induced metric from the word metrics of $\left( G_n ight)_{n\in \mathbb{N}} $ has property A, and the sequence $\left( Q_n ight)_{n\in \mathbb{N}} $ with the quotient metrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence $\left( G_n ight)_{n\in \mathbb{N}}$, 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 \cite{Arzhantseva-Tessera 2015}.

Motivation & Objective

• To establish the coarse Baum-Connes conjecture for sequences of group extensions where the normal subgroup has property A and the quotient group coarsely embeds into Hilbert space.
• To resolve an open problem posed by Arzhantseva and Tessera (2015) concerning relative expanders and box spaces that do not coarsely embed into Hilbert space.
• To extend the applicability of the coarse Baum-Connes conjecture beyond groups with coarse embeddings into Hilbert space.
• To provide a framework for verifying the conjecture in cases where traditional embedding criteria fail.

Proposed method

• Use of the induced word metric on the normal subgroup sequence (N_n) to analyze its geometric properties.
• Assumption that (N_n) has property A, ensuring favorable geometric behavior for K-theoretic computations.
• Assumption that (Q_n) with quotient metrics coarsely embeds into Hilbert space, enabling the use of Hilbert space techniques.
• Application of exact sequences of groups to relate the geometry of (G_n) to those of (N_n) and (Q_n).
• Use of coarse geometry tools to analyze the large-scale structure of (G_n) despite potential lack of Hilbert space embeddability.
• Leveraging known results on property A and coarse embeddability to deduce the coarse Baum-Connes conjecture for (G_n).

Experimental results

Research questions

• RQ1 Can the coarse Baum-Connes conjecture be verified for group extensions when the full group does not coarsely embed into Hilbert space?
• RQ2 Under what conditions on the normal and quotient groups does the coarse Baum-Connes conjecture hold for the extension?
• RQ3 Do relative expanders and certain box spaces of free groups satisfy the coarse Baum-Connes conjecture despite lacking Hilbert space embeddability?
• RQ4 Is property A on the normal subgroup sufficient to ensure the conjecture when combined with coarse embeddability of the quotient?
• RQ5 Can the conjecture be established for groups that do not contain weakly embedded expanders?

Key findings

• The coarse Baum-Connes conjecture holds for the sequence (G_n) under the assumptions that (N_n) has property A and (Q_n) coarsely embeds into Hilbert space.
• The result applies to relative expanders constructed by Arzhantseva and Tessera, which do not coarsely embed into Hilbert space but satisfy the conjecture.
• The method resolves an open problem raised by Arzhantseva and Tessera (2015) regarding the coarse Baum-Connes conjecture for relative expanders.
• The framework extends to special box spaces of free groups introduced by Delabie and Khukhro, which also do not coarsely embed into Hilbert space but satisfy the conjecture.
• The paper demonstrates that the absence of Hilbert space embeddability does not obstruct the coarse Baum-Connes conjecture when geometric conditions on the normal and quotient groups are met.