One of the pinnacle results so far about the (strong) Novikov conjecture is Guoliang Yu’s proof that it holds for groups which are coarsely embeddable into a Hilbert space. In fact, he first proved that under this assumption the coarse Baum-Connes conjecture holds, and then one can invoke the descent principle to get to the strong Novikov conjecture.
Though being coarsely embeddable is an extremely general notion, it is not preserved under extensions as Arzhantseva and Tessera show (arXiv:1605.01192). So the question arises whether either the strong Novikov conjecture or the coarse Baum-Connes conjecture hold for such extensions. The case of the strong Novikov conjecture was affirmatively settled by Jintao Deng (arXiv:1910.05381), so let me focus on the coarse Baum-Connes conjecture.
In the following we need the notion of exactness, which is a bit stronger than being coarsely embeddable. We abbreviate being exact by A (this comes from the notion of ‘Property A’ which is equivalent to exactness) and being coarsely embeddable by CE, and if \(1 \to N \to G \to Q \to 1\) is a short exact sequence we say that it is, e.g., A-by-CE if N is exact and Q is coarsely embeddable.
- If the extension is A-by-A, then G is again exact and hence satisfies the coarse Baum-Connes conjecture. A proof of this may be found in ‘Anantharaman-Delaroche, Renault – Amenable groupoids’ or in a paper of Kirchberg and Wassemann (https://www.emis.de/journals/DMJDMV/vol-04/16.html).
- Similar to the previous point, Dadarlat and Guentner proved that CE-by-A implies CE (https://www.jstor.org/stable/1194929) and hence again the coarse Baum-Connes conjecture.
- The case of A-by-CE extensions is now much more interesting: examples of Arzhantseva-Tessera and of Delabie-Khukhro show that coarse embeddability might fail for groups that are A-by-CE extensions, but on the other hand Deng-Wang-Yu recently showed that such groups still satisfy the coarse Baum-Connes conjecture (arXiv:2102.10617).
- The remaining case of CE-by-CE is currently open, i.e., it is not known if groups defined by such extensions must satisfy the coarse Baum-Connes conjecture.
There are also other coarse geometric questions that one might ask about groups defined by extensions as discussed above. To mention one that interests me, consider the problem whether for a group G the maximal and reduced (uniform) Roe algebras coincide, resp. have isomorphic K-theories:
- Špakula and Willett proved that if G is exact, then its maximal and reduced (uniform) Roe algebras coincide, and if G is coarsely embeddable, then they have isomorphic K-theories (arXiv:1110.1531).
- Since an A-by-A group is again exact and a CE-by-A group is again CE, these cases are directly treated by the result of Špakula and Willett.
- So the remaining question is what happens in the case of A-by-CE, or even of CE-by-CE groups – do their maximal and reduced (uniform) Roe algebras have isomorphic K-theories?
edit (April 20th, 2021): Jintao Deng told me that their proof (the one in arXiv:2102.10617) almost also shows what I ask, i.e., that for A-by-CE groups the maximal and reduced (uniform) Roe algebras have isomorphic K-theories (they are showing it for the twisted Roe algebras). Hence it should be possible to adapt that part of their proof to really show what is asked by me.
edit (November 2nd, 2021): Liang Guo, Zheng Luo, Qin Wang and Yazhou Zhang proved now in detail that for A-by-CE groups the maximal and reduced (uniform) Roe algebras have isomorphic K-theories (arXiv:2110.15624).