arXiv research

We obtain a description of Poisson--Furstenberg boundaries for (random walks on) fundamental groups of compact graph-manifolds. Together with previously known results due to V.A. Kaimanovich and others, this allows one to obtain descriptions of Poisson--Furstenberg boundaries for fundamental groups of all closed 3-mani…

Develops measures for non-Borel Anosov groups on Furstenberg boundary.

We show under weak hypotheses that $\partial X$, the Roller boundary of a finite dimensional CAT(0) cube complex $X$ is the Furstenberg-Poisson boundary of a sufficiently nice random walk on an acting group $Γ$. In particular, we show that if $Γ$ admits a nonelementary proper action on $X$, and $μ$ is a generating prob…

Study shows singularity of stationary measure on Furstenberg boundary for certain random walks.

The study finds discrete subgroups with full limit sets in higher rank Lie groups.

We give elementary constructions for Satake-Furstenberg, Martin and Karpelevich boundaries of symmetric spaces. We also consruct some "new" boundaries

Irreducible groups cannot be free if they ergodically act on a boundary.

We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to …

The paper studies maps and reducibility for cocycles into CAT(0)-spaces.

Projections from flats to maximal flats defined and studied.

We consider the Einstein deformations of the reducible rank two symmetric spaces of noncompact type. If $M$ is the product of any two real, complex, quaternionic or octonionic hyperbolic spaces, we prove that the family of nearby Einstein metrics is parametrized by certain new geometric structures on the Furstenberg bo…

Study on horospheres in higher rank homogeneous spaces, proving density properties.

The torus $\mathbb{T}=S^1\times S^1$ appears as the ideal boundary $\partial_\infty AdS^3$ of the three-dimensional anti-de Sitter space $AdS^3$, as well as the Fürstenberg boundary $\mathbb{F}(X)$ of the rank-2 symmetric space $X={\rm SO}_0(2,2)/{\rm SO}(2)\times{\rm SO}(2)$. We introduce cross-ratios on the torus in …

The paper tackles Kakeya and Nikodym sets on curved manifolds, reducing problems to Euclidean space.

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov…

Study random walks on CAT(0) spaces with contracting elements, proving limit laws.

Using a characterization of parabolics in reductive Lie groups due to Furstenberg, elementary properties of buildings, and some algebraic topology, we give a new proof of Tits' classification of 2-transitive Lie groups.

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…

We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions are homothetic in restriction to the set of primitive elements of $F_N$. We identify the set of Busemann points with the s…

Formula for harmonic current dimension on foliated surfaces, extending Brunella's inequality.

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Suc…

We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization which completely solves the discrete analogue of the Furstenberg set problem in all…

Using optimal transport we study some dynamical properties of expanding circle maps acting on measures by push-forward. Using the definition of the tangent space to the space of measures introduced by Gigli, their derivative at the unique absolutely continuous invariant measure is computed. In particular it is shown th…

Researchers prove hitting measure singularity for most Fuchsian and Kleinian groups.

Geometrically describes Satake compactifications without root data.

Let G be a complex semisimple Lie group, K a maximal compact subgroup and V an irreducible representation of K. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure on M we construct a map from the Satake compactification of G/K (associated to V) to the Lie algebra of …

Study of flows on circle bundles over translation surfaces, showing decay of correlations.

Classifies measures for Anosov subgroups in higher ranks.

Proves equivalence of two types of boundaries in metric spaces.

Proves well-posedness for Einstein equations with specific boundary conditions.

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the …

Study shows boundary curves of free boundary minimal surfaces are circles.

Abstracts a construction of boundary triplets for self-adjoint elliptic problems.

The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonst…

Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.

Generalizes Bestvina's $\mathcal{Z}$-boundaries to coarse $\mathcal{Z}$-boundaries.

We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a prope…

Study on quasi-Einstein manifolds with boundary estimates and inequalities.

Proof of local well-posedness for a specific boundary condition in general relativity.

Foundations for free boundary Brakke flows established.

Embedded surfaces in a ball have any genus and connected boundary.

Study finds rigid properties of boundary-free hypersurfaces in specific data sets.

The paper studies $\mathfrak{g}$-stability of surfaces with boundary and derives area estimates.

The paper shows how sublinearly Morse boundaries can be understood through combinatorial methods.

Homotopy equivalent boundaries of cube complexes are studied.

New examples of non-smoothable homeomorphisms of 4-manifolds with boundary found.

To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed t…