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**On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds.**
*(English)*
Zbl 1316.11042

Let \(X\) be a closed hyperbolic three-manifold with fundamental group \(\Gamma\subset \mathrm{PSL}_2(\mathbb{C})\). Then if \(E_{2k}\) is the flat vector bundle over \(X\) induced by the natural representation of \(\mathrm{PSL}_2(\mathbb{C})\) on the \(2k\)th symmetric power \(V(2k)\) of \(\mathbb{C}^2\) we have \(H^*(X,E_{2k}) = 0\) for all \(k\geq 1\). On the other hand, if \(\Gamma\) stabilizes a lattice \(M_{2k}\) in \(V(2k)\) then we get a local system \(\mathcal{M}_{2k}\) of \(\mathbb{Z}\)-modules on \(X\) and the finite cohomology groups \(H^*(X,\mathcal{M}_{2k}) = H^*(\Gamma,M_{2k})\) are usually nonzero. In fact the second cohomology group can be quite large, and the purpose of the present paper is to analyze the growth of the size of \(H^2(X,\mathcal{M}_{2k})\) as \(k\) goes to infinity. The motivation for this comes mostly from number theory, as the torsion cohomology classes correspond to “automorphic forms mod \(p\)” which have recently been proven to yield Galois representations via a Langlands-type correspondence (see the preprint [P. Scholze, “On torsion in the cohomology of locally symmetric varieties”, Preprint, arxiv:1306.2070). Finally we note that a similar question arises when one varies the manifold \(M\) instead of the local system; the latter was initially considered by N. Bergeron and A. Venkatesh [J. Inst. Math. Jussieu 12, No. 2, 391–447 (2013; Zbl 1266.22013)].

The main result in the paper under review is the following: if \(\Gamma\) is a torsion-free arithmetic group commensurable to the group of units of an order in a division quaternion algebra over an imaginary quadratic number field, then it stabilizes a lattice in each \(V(2k)\) and the limit: \[ \lim_{k\to+\infty} \frac{\log|H^2(X,\mathcal{M}_{2k})|}{k^2} = \frac{2}{\pi}\mathrm{vol}(X) \] holds. The proof relies on the Cheeger-Müller theorem for unimodular bundles proven by the second author in [J. Am. Math. Soc. 6, No. 3, 721–753 (1993; Zbl 0789.58071)] to relate homological torsion to the analytic torsion, and on his work analyzing the growth of analytic torsion in [Prog. Math. 297, 317–352 (2012; Zbl 1264.58026)]. The additional ingredient in the present paper is the analysis of the growth of \(H^p\) for \(p=1,3\), for which it is proven that \(\log|H^p(X,\mathcal{M}_{2k})| \ll k\log k\).

The authors also express the leading coefficient of the Laurent expansion of the Ruelle zeta function associated to the representation of \(\Gamma\) on \(V\) in terms of the integral cohomology \(H^*(X,M)\) if there is a \(\Gamma\)-stable lattice \(M\subset V\). Again, the proof of this is essentially a reformulation of results in the two papers of the second author quoted above.

The main result in the paper under review is the following: if \(\Gamma\) is a torsion-free arithmetic group commensurable to the group of units of an order in a division quaternion algebra over an imaginary quadratic number field, then it stabilizes a lattice in each \(V(2k)\) and the limit: \[ \lim_{k\to+\infty} \frac{\log|H^2(X,\mathcal{M}_{2k})|}{k^2} = \frac{2}{\pi}\mathrm{vol}(X) \] holds. The proof relies on the Cheeger-Müller theorem for unimodular bundles proven by the second author in [J. Am. Math. Soc. 6, No. 3, 721–753 (1993; Zbl 0789.58071)] to relate homological torsion to the analytic torsion, and on his work analyzing the growth of analytic torsion in [Prog. Math. 297, 317–352 (2012; Zbl 1264.58026)]. The additional ingredient in the present paper is the analysis of the growth of \(H^p\) for \(p=1,3\), for which it is proven that \(\log|H^p(X,\mathcal{M}_{2k})| \ll k\log k\).

The authors also express the leading coefficient of the Laurent expansion of the Ruelle zeta function associated to the representation of \(\Gamma\) on \(V\) in terms of the integral cohomology \(H^*(X,M)\) if there is a \(\Gamma\)-stable lattice \(M\subset V\). Again, the proof of this is essentially a reformulation of results in the two papers of the second author quoted above.

Reviewer: Jean Raimbault (Toulouse)

### MSC:

11F75 | Cohomology of arithmetic groups |

22E40 | Discrete subgroups of Lie groups |

58J52 | Determinants and determinant bundles, analytic torsion |

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\textit{S. Marshall} and \textit{W. Müller}, Duke Math. J. 162, No. 5, 863--888 (2013; Zbl 1316.11042)

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