We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
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48 results for “higher-rank Anosov actions”
Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actionsmath.DG
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…
We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
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.
Innovates rotation index for matrix pairs, solving group action problems.
problem Solving group actions problems, especially Nielsen realization and higher-rank Anosov actions.
method Rotation index and Milnor--Munkres--Novikov pairing applied to group actions.
result Solved specific group action problems using new matrix pair invariant.
In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov'…
We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…
The study proves no -eigenvalues for higher rank locally symmetric spaces.
problem Absence of principal eigenvalues for higher rank locally symmetric spaces.
method Derives dynamical assumptions on the Γ-action on geodesics and Satake compactifications.
result Generalization of Patterson's result to higher rank locally symmetric spaces.
The paper classifies fiber structures of discontinuity domains for Anosov representations.
problem Understanding the topology of discontinuity domains for Anosov representations.
method Explicitly working out a smooth version of Fintushel's classification theorem for -actions on 4-manifolds.
result The action on the fiber is equivalent to a circle action on a Hirzebruch surface.
The paper studies ergodicity of flows on subspaces, generalizing earlier work.
problem Ergodicity of flows on subspaces of higher rank groups.
method Analyzes one-parameter diagonalizable subgroups of connected semisimple groups acting on homogeneous spaces.
result Obtains an ergodicity criterion similar to Hopf-Tsuji-Sullivan for general Anosov subgroups.
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.
Anosov groups' measures on limit sets are uniquely determined by their dimension.
problem Characterizing measures on limit sets of Anosov groups.
method Higher rank Hopf-Tsuji-Sullivan dichotomy for maximal diagonal actions.
result Uniqueness of -conformal measures for critical dimensions.
New concept of relatively dominated representations for higher-rank groups.
problem Understanding geometric finiteness in higher-rank Lie groups.
method Introducing and analyzing relatively dominated representations.
result Groups admitting relatively dominated representations are relatively hyperbolic.
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.
Extends Hopf-Tsuji-Sullivan dichotomy to higher rank groups and applies to Anosov subgroups.
problem Understanding discrete subgroups of semisimple real algebraic groups.
method Establishes an extension of the Hopf-Tsuji-Sullivan dichotomy and applies it to Anosov subgroups.
result Anosov subgroups exhibit different phenomena depending on the rank of the group.
Global rigidity theorem for certain lattice actions on manifolds.
problem Volume-preserving actions of higher rank lattices on manifolds with dominated splitting.
method Proves standard conjugacy of actions with dominated splitting.
result Actions must be standard, manifold is flat torus with affine action.
Smooth classification of Cartan actions of higher rank semisimple Lie groups and their latticesmath.DG
Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let Γ\subset G be an irreducible lattice. We provide a C^\infty classification for volume-preserving Cartan actions of Γand G. Also, if G has real rank at least 3, we provide a C^\infty classification for volume-preserv…
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.
In this note we give an overview of some of our recent work on Anosov representations of discrete groups into higher rank semisimple Lie groups.
We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of ran…
Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not a…
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.
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.
Defines new representations for hyperbolic groups, unifying existing definitions.
problem Geometrically finite behavior in higher rank groups.
method Introduces a new family of discrete representations for relatively hyperbolic groups.
result Stability of these representations under certain deformations.
This note removes technical assumptions and characterizes relatively dominated representations.
problem Geometrically finiteness and Anosov conditions in higher-rank settings.
method Characterization using eigenvalue gaps and limit maps.
result Relatively dominated representations are characterized using eigenvalue gaps and limit maps.
We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit…
Study of Anosov representations with Lipschitz limit set and applications to rigidity.
problem Characterizing Anosov representations with specific limit set properties.
method Introducing an unstable Jacobian and analyzing its orbit growth rate.
result Many higher rank representations belong to the studied class.
Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).
problem Classifying Anosov representations of hyperbolic triangle groups into SL(3,R).
method Proving representations are Anosov if they lie in the Hitchin component or the Barbot component, with specific conditions for eigenvalues.
result Anosov representations in SL(3,R) have non-convex boundary maps.
New examples of embeddings defy Anosov representation limits.
problem Examples of robust quasi-isometric embeddings not approximated by Anosov representations.
method Exhibited non-locally rigid, Zariski dense embeddings in .
result Higher rank Anosov representation theorems fail for .
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.
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…
Uniformizes compact complex manifolds via Anosov representations.
problem Uniformization of compact complex manifolds.
method Anosov homomorphisms with small limit sets.
result Local homeomorphism of character variety to Teichmüller space.
CAT(0) spaces of higher rankmath.MG
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 are rigid if they contain a periodic -flat.
The paper constructs Anosov representations for specific types of groups.
problem Constructing Anosov representations for certain groups.
method Analyzing uniform lattices and their extensions, proving existence of Anosov embeddings.
result Examples of one-ended hyperbolic groups admit Anosov embeddings into higher-rank Lie groups.
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…
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.
Study on horospheres in higher rank homogeneous spaces, proving density properties.
problem Density of horospheres in higher rank homogeneous spaces.
method Analyzing maximal horospherical subgroups and their minimal subsets in the context of Furstenberg boundary.
result Equivalence of horospherical limit points and density properties in higher rank homogeneous spaces.
Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigiditymath.GT
The paper introduces Patterson-Sullivan systems and proves their rigidity, with applications to random walks and entropy rigidity.
problem Understanding the rigidity of Patterson-Sullivan systems and their applications.
method Generalization of Tukia's measurable boundary rigidity theorem for Patterson-Sullivan systems.
result Entropy rigidity for Anosov groups with Lipschitz limit sets.
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.
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 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 are conjugate to affine actions on (infra-)tori.
The paper counts conjugacy classes of loxodromic elements in Anosov subgroups with a power saving error term.
problem Counting conjugacy classes of loxodromic elements in Anosov subgroups.
method Interpreting Jordan projections as periods of a flow and proving exponential mixing.
result Proves a counting theorem with a power saving error term for conjugacy classes of loxodromic elements.
The study extends Dehn filling to Lie groups, ensuring geometric properties.
problem Generalizing Dehn filling to semisimple Lie groups.
method Analyzing deformations of subgroups and their geometric properties.
result Extended geometrically finite subgroups can be deformed while maintaining properties.
Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
New subgroup found in Lie groups with unusual properties.
problem Finding discrete subgroups with specific properties in Lie groups.
method Constructing a specific subgroup of a higher rank Lie group.
result Found a new subgroup that is dense, discrete, non-lattice, and non-tempered.
Study shows how to detect representation extendability using conformal measures.
problem Detecting extendability of representations using conformal measures.
method Using higher rank conformal measures and self-joinings of groups.
result Affirmative answer to detect extendability of representations.
New insights into the geometry of flows on 3-manifolds.
problem Understanding the geometry of flows on 3-manifolds.
method Analyzing the action of pseudo-Anosov flows on Gromov-hyperbolic spaces.
result Genericity of non-periodic elements in the fundamental group.
Study non-transitive pseudo-Anosov flows using group actions.
problem Characterize pseudo-Anosov flows in 3-manifolds.
method Extend pseudo-Anosov action to non-transitive flows, use group actions on orbit spaces and boundary at infinity.
result Pseudo-Anosov flows in 3-manifolds are determined by their group actions on boundary at infinity.