Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
The paper encourages Kleinian group thinking for higher rank Lie groups.
problem No specific problem stated; encouraging new thinking.
method Discussion of Kleinian group ideas applied to higher rank Lie groups.
result Encouragement to think about higher rank Lie groups using Kleinian group theory.
We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank latti…
Study higher rank inner products and their tilings to describe tori degenerations.
problem Understanding metric degenerations of tori.
method Introduce higher rank inner products and their tilings, use to describe degenerations.
result Describe metric degenerations of polarized tori and Hausdorff limits of tilings.
Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.
problem CAT(0) spaces of higher rank with geometric group actions.
method Proving rigidity for spaces containing periodic flats and geodesics in flats.
result CAT(0) spaces of higher rank n≥2 are rigid if they contain a periodic n-flat. Characterizes higher rank model geometries using antipodal sets.
problem Identifying higher rank model geometries among Hadamard spaces.
method Using antipodal sets at infinity to characterize model geometries.
result Characterizes Riemannian symmetric spaces, Euclidean buildings, and products as higher rank model geometries.
The study proves conditions for CAT(0) spaces with higher rank rigidity.
problem Conditions for rigidity in CAT(0) spaces with higher rank.
method Geometric group action and analysis of geodesics and flats.
result CAT(0) spaces with specific properties are classified.
We show that polar actions of cohomogeneity two on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove that polar actions induced by reductive algebraic subgroups in the isometry grou…
The paper proves actions of lattices in higher rank groups have cost one.
problem Fixed price question for higher rank semisimple Lie groups.
method Low intensity Poisson point processes and geometry of Voronoi tessellations.
result Proves all probability measure preserving actions of lattices in higher rank groups have cost one.
We give very flexible, concrete constructions of discrete and faithful epresentations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we find a general criterion for when discrete and faithful representations exist, and…
Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.
problem Investigate critical exponents for vanishing L^p-cohomology in higher rank Lie groups and manifolds.
method Examine SL3(R) and 5-dimensional solvable Lie groups, use spectral sequence arguments. result Discover a continuum of quasi-isometry classes of rank 2 solvable Lie groups.
New proof for higher rank subvarieties in genus three.
problem Classifying higher rank invariant subvarieties in genus three.
method Uses recent techniques developed by Apisa and Wright.
result Short and simplified proof of classification.
New dHYM connections found on complex vector bundles.
problem Existence of dHYM connections on higher rank vector bundles.
method Constructing explicit non-trivial examples and providing algebraic conditions.
result First explicit non-trivial dHYM connections on higher rank holomorphic vector bundles.
For convex real projective manifolds we prove an analogue of the higher rank rigidity theorem of Ballmann and Burns-Spatzier.
We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.
Improved homological dimension for certain subgroups in Lie groups.
problem Determining homological dimensions of discrete subgroups in Lie groups.
method Using recent results and properties of injectivity radius, the homological dimension gap is calculated.
result Infinite volume torsion-free subgroups of higher rank Lie groups have a homological dimension gap of at least 1/8 of the real rank.
The paper proves that certain spaces have injective balls of any radius.
problem The injectivity radius of certain geometric spaces is infinite.
method Analyzes higher rank simple and semisimple Lie groups with specific properties.
result The locally symmetric spaces have injective balls of any radius.
Researchers show how to perturb free group representations into higher rank groups.
problem Understanding representations of free groups into non-Anosov Lie groups.
method Study quasi-isometric representations and show perturbations.
result Some representations can be perturbed to be non-quasi-isometric or unstable.
New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse r-median spaces and proving their existence. result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
The paper proves rigidity and ergodicity of horospherical foliations.
problem Rigidity and ergodicity of horospherical foliations in higher rank.
method Establishes higher rank extensions of rigidity theorems for representations of discrete subgroups of divergence type, using conformal measures and boundary maps.
result Proves conformal measure rigidity and ergodicity of horospherical foliations for hypertransverse subgroups.
The study finds discrete subgroups with full limit sets in higher rank Lie groups.
problem Finding discrete subgroups with full limit sets in higher rank Lie groups.
method Analyzing real semi-simple Lie groups of higher rank and providing criteria for discrete subgroups of G=SL(3,R). result Existence of discrete subgroups with full limit sets in higher rank Lie groups.
In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion. The main result of this paper is ME r…
We classify all holomorphic actions of higher rank lattices on compact Kaehler manifolds of dimension 3. This provides a complete answer to Zimmer's program for holomorphic actions on compact Kaehler manifolds of dimension at most 3.
The prime geodesic theorem for regular geodesics in a higher rank locally symmetric space is proved. An application to class numbers is given. The proof relies on a Lefschetz formula that is based on work of Andreas Juhl.
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For sy…
Extends confining subset theory to describe hyperbolic actions of solvable groups with higher rank abelianizations.
problem Describing hyperbolic actions of solvable groups with higher rank abelianizations.
method Extends confining subset theory to apply to solvable groups with higher rank abelianizations.
result Complete description of hyperbolic actions of generalized solvable Baumslag-Solitar groups.
Study critical exponents in normal subgroups of higher rank Lie groups.
problem Understanding critical exponents in normal subgroups of higher rank Lie groups.
method Analyzing subgroups and their critical exponents in a higher rank semi-simple Lie group.
result Critical exponents of normal subgroups coincide under certain conditions.
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
Classifies measures for Anosov subgroups in higher ranks.
problem Classifying horospherical invariant measures for Anosov subgroups.
method Geometric approach, not relying on flows or ergodic theorems.
result Extends results from rank one to higher ranks, solving open problems.
The aim of this paper is to study the spectrum of the Lp Laplacian and the dynamics of the Lp heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in t…
Witten's conjecture suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. A higher rank version of the Donaldson invariants was introduced by Kronheimer. Before even having been defined, the physicists Mariño and…
We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.
The paper proves rigidity for cocycles from higher rank lattices to Out(FN).
problem Proving rigidity for cocycles from higher rank lattices to Out(FN).
method Geometric tool: barycenter map.
result Every Borel cocycle is cohomologous to a cocycle with finite image.
In this paper we study a proposal of Nekrasov, Rosly and Shatashvili that describes the effective twisted superpotential obtained from a class S theory geometrically as a generating function in terms of certain complexified length-twist coordinates, and extend it to higher rank. First, we introduce a higher rank analog…
Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduc…
In this note, we present a new way to associate a spectral triple to the noncommutative C∗-algebra C∗(Λ) of a strongly connected finite higher-rank graph Λ. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph C∗-algebras C∗(Λ), and we prove that these s…
Research examines coamenable subgroups in higher rank groups.
problem Investigates coamenable normal subgroups in higher rank groups.
method Analyzes three complementary phenomena in higher rank groups.
result Growth indicators of coamenable subgroups are not preserved but the Riemannian critical exponent remains rigid.
The paper describes correlations of spectra for higher rank Anosov representations.
problem Understanding correlations of spectra for Anosov representations of higher rank groups.
method Relates correlation problem to counting projections in truncated hypertubes.
result Extends previous work on rank one representations to higher rank.
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…
Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.
problem Establishing existence and uniqueness of Patterson-Sullivan measures in higher rank symmetric spaces.
method Develops theory for vector-valued horofunction boundaries and shadows.
result Proves existence and uniqueness of Patterson-Sullivan measures for transverse groups.
We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded nonpositive flag curvature with rank at least 2 whose universal cover is irreducible, is a locally symmetric space or a locall…
Proves finite measure implies product structure for certain discrete subgroups.
problem Classifying discrete subgroups with finite Bowen-Margulis-Sullivan measure.
method Product structure of leafwise measures and high entropy method.
result Proves virtually a product structure for certain subgroups.
Constructs geometries with nonvanishing curvature and essential automorphisms.
problem Creating geometries with nonvanishing curvature and essential automorphisms.
method Using elements of the kernel of the Kostant Laplacian to construct homogeneous Cartan geometries, then modifying them to make base manifolds compact.
result Infinite families of regular normal Cartan geometries with nonvanishing curvature and essential automorphisms on closed manifolds for higher rank parabolic model geometries.
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. Paper proves finite BMS measure for SPR groups in higher rank Lie groups.
problem Finite measure for certain groups in higher rank Lie groups.
method Developed SPR property and proved finite BMS measure.
result Finite Bowen-Margulis-Sullivan measure for SPR groups in higher rank Lie groups.
Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.
problem Positive entropy actions by higher-rank lattices in Lie groups.
method Analysis of sub-actions, fiber entropy upper semicontinuity, and conjugacy arguments.
result Actions by higher-rank lattices in SL(n,R) are conjugate to affine actions on (infra-)tori. We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of…