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.
We use Luttinger surgery to show that there are no Lagrangian Klein bottles in S^2× 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.
This paper builds one-cusped complex hyperbolic 2-manifolds by an explicit geometric construction. Specifically, for each odd d >= 1 there is a smooth projective surface Z(d) with .Zd / Dc(1)(2)(Z(d)) =c(2)(Z(d))= 6d and a smooth irreducible curve E(d )on Z(d) of genus 1 so that Z(d) X E-d admits a finite volume uniformization by the unit ball B- 2 in C-2 . This produces one-cusped complex hyperbolic 2-manifolds of arbitrarily large volume. As a consequence, the 3-dimensional nilmanifold of Euler number 12d bounds geometrically for all odd d >= 1
Let f be a complex Hénon map and μ its unique measure of maximal entropy. We prove that μ is exponentially mixing of all orders for all (not necessarily bounded) plurisubharmonic observables, and that all plurisubharmonic functions satisfy the central limit theorem with respect to μ. Our results hold more generally for every Hénon-Sibony map on ℂ^k.
We show that there exists an inscribed square in a Jordan curve given as the union of two graphs of functions of Lipschitz constant less than 1 + root 2. We are motivated by Tao's result that there exists such a square in the case of Lipschitz constant less than 1. In the case of Lipschitz constant 1, we show that the Jordan curve inscribes rectangles of every similarity class. Our approach involves analyzing the change in the spectral invariants of the Jordan Floer homology under perturbations of the Jordan curve.
We show that the Tate conjecture for divisors over a finite field F is equivalent to an explicit algebraic problem about the third Milnor K-group of the function field F(x,y,z) in three variables over F.
We prove that for a homeomorphism f that is isotopic to the identity on a closed hyperbolic surface, the following are equivalent: * f acts hyperbolically on the fine curve graph; * f is isotopic to a pseudo-Anosov map relative to a finite f-invariant set; * the ergodic homological rotation set of f has nonempty interior.
We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $\mu$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $\mu$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,\mu)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $\mu$ with a finite first moment.
We introduce a natural way of associating oriented closed geodesics on the modular curve to elements of $(\mathbb{Z}/q\mathbb{Z})^\times$ and prove that the corresponding packets associated to sufficiently large subgroups equidistribute in the unit tangent bundle as $q$ tends to infinity. This is a $q$-orbit analogue of Duke's Theorem for real quadratic field as extended to subgroups by Popa. We also show that the homology classes of the $q$-orbits of oriented closed geodesics concentrate around the Eisenstein line and present group theoretic applications thereof.
In this note we prove Sarnak's (spherical) density hypothesis for the full discrete spectrum of the quotients $\Gamma_{\textrm{pa}}(q)\backslash \textrm{Sp}_4(\mathbb{R})$, where $\Gamma_{\textrm{pa}}(q)$ are paramodular groups with square-free level $q$. To derive this estimate we upgrade a density estimate established via the (pre)-Kuznetsov formula, which only accounts for the generic part, using Arthur's parametrization of the discrete spectrum.
We prove a version of Linnik's basic lemma uniformly over the base field using theta-series and geometric invariant theory in the spirit of Khayutin's approach (Duke Math. J., 168(12), 2019). As an application, we establish entropy bounds for limits of invariant measures on homogeneous toral sets in GL(4) of biquadratic, cyclic, or dihedral type.
We describe the dynamics of a group \Gamma generated by Dehn twists along two filling multi-curves or a family of filling curves on the \mathsf{SU}(2) -representation variety of closed surfaces. Consequently, we provide explicit \Gamma -invariant rational functions on the representation variety of the genus two closed surface S_{2} for some pair of multi-curves. We establish a similar result for the \mathsf{SU}(2) -character variety of genus four non-orientable surfaces N_{4} for some family of filling curves.
Morita [Osaka J. Math. 21 (1984), 545-563] showed that for each integer k >= 1 there are examples of flat S-1-bundles for which the k-th power of the Euler class does not vanish. Haefliger | Enseign. Math. (2) 24 (1978), 154] asked if the same holds for flat odd-dimensional sphere bundles. In this paper, for a manifold M with a free torus action, we prove that certain M-bundles are cobordant to a flat M-bundle and as a consequence, we answer Haeffiger's question. We show that all monomials in the Euler class and Pontryagin classes p(i) for i <= n - 1 are non-trivial in //* (BDiff(+)(delta) (S2n-1); Q). +
In their 2021 and 2022 papers, Cristofaro-Gardiner, Humilière, Mak, Seyfaddini, and Smith defined links spectral invariants on connected compact surfaces and used them to show various results on the algebraic structure of the group of area-preserving homeomorphisms of surfaces, particularly in cases where the surfaces have genus zero. We show that on surfaces with higher genus, for a certain class of links, the invariants will satisfy a local quasimorphism property. Subsequently, we generalize their results to surfaces of any genus. This extension includes the non-simplicity of (i) the group of hameomorphisms of a closed surface, and (ii) the kernel of the Calabi homomorphism inside the group of hameomorphisms of a surface with non-empty boundary. Moreover, we prove that the Calabi homomorphism extends (non-canonically) to the C^{0} -closure of the set of Hamiltonian diffeomorphisms of any surface. The local quasimorphism property is a consequence of a quantitative Künneth formula for a connected sum in Heegaard–Floer homology, inspired by the results of Ozsváth and Szabó.
We introduce the concept of non-Archimedean metrics attached to a transcendental pseudoeffective cohomology class on a compact K\"ahler manifold. This is obtained via extending the Ross-Witt Nystr\"om correspondence to the relative case, and we point out that our construction agrees with that of Boucksom-Jonsson when the class is induced by a pseudoeffective $\mathbb Q$-line bundle. We introduce the notion of a flag configuration attached to a transcendental big class, recovering the notion of a test configuration in the ample case. We show that non-Archimedean finite energy metrics are approximable by flag configurations, and very general versions of the radial Ding energy are continuous, a novel result even in the ample case. As applications, we characterize the delta invariant as the Ding semistability threshold of flag configurations and filtrations and prove a YTD type existence theorem in terms of flag configurations.
For a closed orientable irreducible $3$-manifold $M$ that admits a co-orientable taut foliation with one-sided branching, we show that $\pi_1(M)$ is left orderable.
