Local X-ray Transform on Asymptotically Hyperbolic Manifolds via Projective Compactification

We prove local injectivity near a boundary point for the geodesic X-ray transform for an asymptotically hyperbolic metric even mod $O(\rho^5)$ in dimensions three and higher.

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS

We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math.192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math.125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

LS-category of moment-angle manifolds and higher order Massey products

Using the combinatorics of the underlying simplicial complex K, we give various upper and lower bounds for the Lusternik–Schnirelmann (LS) category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} . We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS-category. In particular, we characterize the LS-category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} over triangulated d-manifolds K for d ≤ 2 {d\leq 2} , as well as higher-dimensional spheres built up via connected sum, join, and vertex doubling operations. We show that the LS-category closely relates to vanishing of Massey products in H * ⁢ ( 𝒵 K ) {H^{*}(\mathcal{Z}_{K})} , and through this connection we describe first structural properties of Massey products in moment-angle manifolds. Some of the further applications include calculations of the LS-category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and k-neighborly complexes, which double as important examples of hyperbolic manifolds.

Ruelle Zeta Functions of Hyperbolic Manifolds and Reidemeister Torsion

This paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at $$s=0$$ s = 0 and its value at $$s=0$$ s = 0 equals the Reidemeister torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried’s result to arbitrary finite dimensional representations of the fundamental group. The Reidemeister torsion is replaced by the complex-valued combinatorial torsion introduced by Cappell and Miller.

A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

We study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in terms of their Gromov boundary, improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition to have isoperimetric inequality and, therefore, it can be removed as hypothesis.