In this note, we prove Sarnak’s (spherical) density hypothesis for the full discrete spectrum of the quotients \Gamma_{\mathrm{pa}}(q) \backslash \operatorname{Sp}_{4}(\mathbb{R}) , where \Gamma_{\mathrm{pa}}(q) are paramodular groups with square-free level q . To derive this estimate we upgrade a density estimate established via the Kuznetsov formula, which only accounts for the generic part, using Arthur’s parametrization of the discrete spectrum.
We prove that curves of constant torsion satisfy the C-1-dense h-principle in the space of immersed curves in Euclidean space. In particular, there exists a knot of constant torsion in each isotopy class. Our methods, which involve convex integration and degree theory, quickly establish these results for curves of constant curvature as well.
We use Luttinger surgery to show that there are no Lagrangian Klein bottles in S-2 x S(2 )in the Z(2)-homology class of an S-2-factor if the symplectic area of that factor is at least twice that of the other.
We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0) space. This provides a partial answer to an old question in geometric group theory and proves partly a conjecture formulated by Norin-Osajda-Przytycki (2022). It applies in particular to Grigorchuk's groups of intermediate growth and other branch groups as well as to simple groups with the Liouville property such as those found by Matte Bon and by Nekrashevych. The method of proof uses ultralimits, equivariant harmonic maps, subharmonic functions, horofunctions and random walks.
We determine the distribution of the number of saddle connections on a random translation surface of large genus. More specifically, for genus g tending to infinity, the number of saddle connections with lengths in a given interval [a/g, b/g] converges in distribution to a Poisson distributed random variable. Furthermore, the numbers of saddle connections associated to disjoint intervals of lengths are independent.
