Analytic torsion convergence conjecture for Benjamini–Schramm covers

Let MM be a fixed closed hyperbolic 33-manifold, and let MnM_n be covers of MM that Benjamini–Schramm converge to hyperbolic 33-space H3\mathbb{H}^3. The analytic torsion convergence conjecture. One should have

τ(Mn)vol(Mn)16π.\frac{\tau(M_n)}{\operatorname{vol}(M_n)} \to \frac{1}{6\pi}.

This is presented as a broadening of the first, analytic-torsion part of the Bergeron–Venkatesh conjecture from arithmetic towers to all covers of a fixed closed hyperbolic 33-manifold. The source reports computational evidence supporting it, while also giving examples showing that Benjamini–Schramm convergence alone does not force the same conclusion for arbitrary sequences.

Sources & referencesView supporting material

Primary source

Jeffrey F. Brock and Nathan M. Dunfield, “Injectivity radii of hyperbolic integer homology 3-spheres”, arXiv:1304.0391 (2014).

Additional references

2 papers in this index state this conjecture (2012–2013). The statement above is taken from the most recent of them; the others are arXiv:1210.2961.

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