Related papers: Stationary random subgroups in negative curvature

We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity…

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's…

We investigate the relationship between the metric boundary and the Gromov boundary of a hyperbolic metric space. We show that the Gromov boundary is a quotient topological space of the metric boundary, and that therefore a word-hyperbolic…

This paper is devoted to study stability of Lyapunov exponents and simplicity of Lyapunov spectrum for bounded random compact operators on a separable infinite-dimensional Hilbert space from a generic point of view generated by the…

Let $(M,g)$ be a compact Riemannian surface with nonpositive sectional curvature and let $\gamma$ be a closed geodesic in $M$. And let $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator $\Delta_g$ with…

Let $\Gamma'<\Gamma$ be two discrete groups acting properly by isometries on a Gromov-hyperbolic space $X$. We prove that their critical exponents coincide if and only if $\Gamma'$ is co-amenable in $\Gamma$, under the assumption that the…

For a class of asymptotically hyperbolic manifolds, we show that the bottom of the continuous spectrum of the Laplace-Beltrami operator is not an eigenvalue. Our approach only uses properties of the operator near infinity and, in…

We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called…

We study the Lp-properties of positive Rockland operators and define Sobolev spaces on general graded groups. This generalises the case of sub-Laplacians on stratified groups studied by G. Folland in [3]. We show that the defined Sobolev…

Let $\Gamma$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar…

We show that for a pointed Gromov-Hausdorff limit of non-collapsed Riemannian manifolds with bounded Ricci curvature, the local $b_1$ of the regular loci vanishes. We also discuss applications and some open questions.

We consider the reducibility problem of cocycles by isometries of Gromov hyperbolic metric spaces in the Livsic setting. We show that provided that the boundary cocycle (that acts on a compact space) is reducible in a suitable H\"older…

In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or…

We study graphs with nonnegative Bakry-\'Emery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov-Hausdorff convergence we prove that the space of bounded harmonic functions is…

We show that for any finitely generated non-amenable group and any $\varepsilon>0$, there exists some finite symmetric generating set with spectral radius less than $\varepsilon$. We give applications to percolation theory and the theory of…

We prove a finiteness result for the systolic area of groups, answering a question of M. Gromov. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of~2-complexes whose systolic area is uniformly…

A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose…

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the…

Verifying a conjecture of Gromov we establish a generalized Margulis Lemma for manifolds with lower Ricci curvature bound. Among the various applications are finiteness results for fundamental groups of compact $n$-manifolds with upper…

We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends results of Mundet i…