## Number theory and 3-manifolds

It used to be the case that the Langlands programme could be used to say something interesting about arithmetic 3-manifolds qua hyperbolic manifolds. Now, after the work of Agol, Wise, and others has blown the subject to smithereens, this gravy train appears to be over. It seems to me, however, that the great advance in our knowledge of hyperbolic 3-manifolds has precious little to say about arithmetic 3-manifolds qua lattices in semi-simple groups. As a basic example, suppose that $X$ is a maximal compact arithmetic three orbifold associated to a quaternion algebra $Q/F$ for some field $F$ (with the appropriate behavior at the infinite primes). Then one may ask whether $X$ has positive Betti number after some finite congruence cover $\widetilde{X} \rightarrow X$. Let’s call this the virtual congruence positive Betti number conjecture. (This conjecture should be true – it is a consequence of Langland’s conjectural base change for $\mathrm{SL}(2)$, which everyone believes but is probably very difficult.) AFAIK, there’s not really much one can say about this problem from the geometric group theory/RAAG/LERF/etc perspective, where the arithmetic structure of the tautological $\mathrm{SL}(2)$-representation does not seem to play so much of a role. A related question is the extent to which arithmetic 3-manifolds are intrinsically different from their non-arithmetic hyperbolic brethren. Is the virtual congruence Betti number conjecture (for arithmetic manifolds) something that could plausibly answered using geometric group theory?

• Any such cohomology classes will generate (in a congruence tower at some other prime $p$) an admissible $\mathrm{G}(\mathbf{Q}_p)$-representation. Such representations are either infinite dimensional or trivial, and in the latter case the cohomology had to come from $\mathrm{G}(\mathbf{R})$-invariant forms, and so already come from a congruence subgroup. So certainly any new cohomology in a non-congruence cover generates more such cohomology. But yes, I agree, it’s hard to see this having any relation to the cohomology downstairs.