There are many versions of the Baum-Connes conjecture (the original, coarse, with coefficients, etc.). I would like to know what group theory results are needed in order to prove or disprove one of these conjectures.
[Math] Baum-Connes conjecture
reference-request
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The Generalized Riemann Hypothesis (GRH) influences Complexity Theory. In particular, Pascal Koiran proved that the truth of the GRH implies that the problem of "whether a set of polynomial equations has a solution over the complex numbers" is in the second level of the polynomial hierarchy, in the Arthur-Merlin class. I do not know if this statement could be proven without GRH. Thus, truth of the GRH bridges between complexity theory over the complex numbers (continuous), and complexity theory modulo many primes (discrete). Koiran's result is discussed on R.J. Lipton's blog.
One significant application of Koiran's theorem was Greg Kuperberg's lovely and surprising proof that, assuming GRH, it is NP to tell whether a knot is knotted. This result is discussed in this Combinatorics and More blog post. Ian Agol also has a GRH-free proof of the same statement using normal surface theory.
This problem seems to lie quite deep. To illustrate this consider the problem of estimating the least quadratic non-residue $\pmod p$ for a prime $p$ -- obviously if the numbers up to some point generate $({\Bbb Z}/p{\Bbb Z})^*$ then they must contain a quadratic non-residue, so this problem should be ``easier." Vinogradov conjectured that the least quadratic non-residue is $\ll p^{\epsilon}$ for any $\epsilon >0$ but this remains unknown. The best known result is that the least quadratic non-residue is $\ll p^{1/(4\sqrt{e})}$, which follows from Burgess's bound on character sums plus a multiplicative trick of Vinogradov. So unconditionally, one should not expect a result better than $n^{1/(4\sqrt{e})+\epsilon}$ for generating all of $({\Bbb Z}/n{\Bbb Z})^*$. And a result of Harman shows that the numbers up to $n^{1/(4\sqrt{e})+\epsilon}$ do in fact generate all of $({\Bbb Z}/n{\Bbb Z})^*$; see Theorem 3 of his paper.
More remarks: To prove a bound of $(\log p)^{A}$ for the least quadratic non-residue seems to require some version of a quasi Riemann hypothesis: no zeros of Dirichlet $L$-functions to the right of $1-1/A$. A precise relation between zero-free regions and the least quadratic non-residue was established first by Rodosskii. One way of thinking about the group generated by the numbers below $y$ in $({\Bbb Z}/n{\Bbb Z})^*$ is to consider the distribution of $y$-smooth numbers below $x$ in arithmetic progressions $\pmod n$ -- this is the focus of Harman's paper referenced already, and for more work in this direction see Soundararajan and Harper. The ideas here would give more relations between zero-free regions and how big $y$ has to be to generate $({\Bbb Z}/n{\Bbb Z})^*$ (along the lines of Rodosskii's work, which you can find referenced there).
Best Answer
This January in New Orleans Paul Baum gave a rather extensive survey of the history and status of Baum/Connes, so I am guessing that he has a historical survey written up, or quasi-written up, since I am not seeing it on his web page), so I would strongly suggest just asking him. Sadly, I don't believe he is a MO participant.