Equivalent trace sets for arithmetic Fuchsian groups (Q2832842)

An important invariant of a closed hyperbolic surface (or more generally a hyperbolic manifold or more generally still a Riemannian manifold) is its Laplace spectrum (the eigenvalues of the Laplace-Beltrami operator on square-integrable functions, with their multiplicities), which determines the behaviour of various PDEs on it such as the wave or heat equation. For hyperbolic surfaces the Selberg trace formula shows that knowledge of the Laplace spectrum is equivalent to that of its length spectrum, the collection of all lengths of closed geodesic curves on the surface, again with their multiplicities (this straightforward relation does not hold in more general settings).NEWLINENEWLINEA popular topic of research has been to find sets of surfaces (or manifolds) with the same spectrum, but which are pairwise not isomorphic. In particular, such isospectral hyperbolic surfaces exist via various very explicit constructions (a survey of the general problem is given in [\textit{C. S. Gordon}, in: Handbook of differential geometry. Vol. I. Amsterdam: North-Holland. 747--778 (2000; Zbl 0959.58039)]). On the other hand such sets must be finite and contain only surfaces with the same volume. The object of the paper under review is to study the related question where the multiplicities are forgotten, on the length side (which has a priori no clear relation to the Laplace eigenvalues). The length set \(L(M)\) is thus defined as simply the set of lengths of all closed geodesics; in the case of hyperbolic surfaces it is equivalent to know the set of all traces of the associated Fuchsian group. In opposition to isospectral manifolds which have the same volume, manifolds \(M, N\) such that \(L(M)=L(N)\) can have \(\mathrm{vol}(M) / \mathrm{vol}(N)\) arbitrarily large: such examples are obtained in [\textit{C. J. Leininger} et al., Int. Math. Res. Not. 2007, No. 24, Article ID rnm135, 24 p. (2007; Zbl 1158.53032)], as covers of a fixed hyperbolic manifold.NEWLINENEWLINEThe main result in the paper in this regard is to construct infinite sets of noncompact hyperbolic surfaces (and 3-manifolds) of finite volume all having the same length set, and unbounded volumes. This is done in an arithmetic setting, by finding infinite-index subgroups of the modular group \(\mathrm{PSL}_2(\mathbb Z)\) with the same trace set. By known residual properties of Fuchsian groups (LERF) one then deduces that there are infinitely many finite-index subgroups between the full modular group and this subgroup, which in particular have the same trace set as both. The result also applies to certain torsion-free finite index subgroups, hence the geometric results on surfaces. Finally, similar results are obtained for 3-dimensional groups, in particular Bianchi groups such as \(\mathrm{PSL}_2(\mathbb Z[\sqrt{-1}])\).