## 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?

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### 2 Responses to Number theory and 3-manifolds

1. ianagol says:

I doubt it. The constructions carried out to get a cover with positive betti number are likely highly non-congruence. Maybe one could hope to promote virtual positive b_1 in a non-congruence cover to a congruence cover? If one has a tower of congruence covers, then there is a finite-sheeted cover of bounded index of each of these (in a compatible tower) which has positive b_1 (by intersecting with the non-congruence cover with positive b_1). Maybe the rank of b_1 of these covers grows unbounded? One could hope to show that this implies that the b_1 is positive downstairs eventually. But the 1-forms may be highly non-automorphic, in some sense.

• 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.