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Torsion homology growth and cycle complexity of arithmetic manifolds. (English) Zbl 1351.11031

Let \(M_0\) be a three-dimensional hyperbolic congruence manifold, i.e., coming from a congruence subgroup in some arithmetic group. The authors discuss the following conjecture: There is a constant \(C\) such that for every congruence manifold \(M\) of volume \(V\) which covers \(M_0\) the second homology \(H_2(M,\mathbb Z)\) is generated by classes of immersed surfaces of genus less than \(V^C.\)
This question is closely related to studying torsion in the homology of the covering manifolds, which has been taken up by the first and third author in [J. Inst. Math. Jussieu 12, No. 2, 391–447 (2013; Zbl 1266.22013)]. The authors in particular show, that – given two further conditions – the truth of the conjecture would imply that for a sequence of covering congruence manifolds \(M_i\to M_0\) the fraction \(\frac{\log \#H_1(M_i,\mathbb Z)_{\mathrm{tors}}}{\mathrm{vol}(M_i)}\) converges to \(\frac 1{6\pi}\) if the volumes of the \(M_i\) go to infinity. The two further assumptions are precise versions of the wish that there are no small Laplace-eigenvalues on the \(M_i\) and that the first Betti-number of \(M_i\) remains small compared to \(\text{vol}(M_i).\)
One of the main ingredients is the Cheeger-Müller-Theorem which here can be rewritten in a manner relating the order of the torsion-part in \(H_1(M,\mathbb Z)\) and certain regulators, as well as the analytic torsion of \(M.\)
The arithmeticity condition implies the possibility to use Hecke-operators in order to prevent cycles from being too wild.
After the proof of the main theorem, the authors spend more than half of the paper in exhibiting two classes of examples where the conjecture is satisfied. For both types of examples the arithmetic groups have to satisfy further conditions of arithmetic and topological nature, and in this part of the paper the equivariant Birch-Swinnerton-Dyer conjecture or base-change arguments enter into the picture. In a last section, some numerical examples are computed.
The paper is very well written. There is a large amount of topological and automorphic ingredients which the reader is guided through in a careful way.

MSC:

11F75 Cohomology of arithmetic groups
57M50 General geometric structures on low-dimensional manifolds

Citations:

Zbl 1266.22013

Software:

LMFDB
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References:

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