Recent work on Anosov representations of discrete groups.
problem Understanding Anosov representations of discrete groups into higher rank semisimple Lie groups.
method Overview of recent research findings.
result New insights into Anosov representations.
Study primitive stable representations in higher rank Lie groups and their properties.
problem Understanding primitive stable representations in higher rank semisimple Lie groups.
method Analyzing convex projective structures and positive representations on surfaces.
result Holonomies of convex projective structures and positive representations on surfaces with one boundary component are primitive stable.
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.
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.
The paper explores higher property T in lattices and its connections to geometric phenomena.
problem Understanding higher property T in lattices and related geometric phenomena.
method Operator-algebraic characterizations of higher property T and connections to lattice geometry.
result Unified framework for understanding higher property T and related geometric phenomena.
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.
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…
We show that S-arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.
This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…
Proof of boundedness of quasimorphisms for certain Lie groups.
problem Bounding quasimorphisms on lattices in Lie groups.
method Thermodynamic formalism applied to bounded cohomology.
result Every π-quasimorphism on irreducible uniform lattices is bounded.
Study infinite subgroups of higher rank Lie groups, focusing on Anosov subgroups.
problem Understanding properties of Anosov subgroups in higher rank semisimple Lie groups.
method Characterize Anosov subgroups through geometric, coarse geometric, and dynamical viewpoints.
result New equivalent characterizations of Anosov subgroups, capturing rank one behavior.
If G is a semisimple Lie group of real rank at least 2 and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.
Study quasi-isometric embeddings in symmetric spaces and Lie groups.
problem Understanding embeddings between symmetric spaces and Lie groups.
method Decompose embeddings into irreducible components and analyze examples.
result Rigidity results extended to semisimple Lie groups, including counterexamples.
We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.
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…
Let Γ be an irreducible lattice of $\Q$-rank ≥2 in a semisimple Lie group of noncompact type. We prove that any action of Γ on a $\CAT(0)$ cubical complex has a global fixed point.
New theory extends classical results to Anosov subgroups.
problem Classical Patterson-Sullivan theory applied to Anosov subgroups.
method Invariant Finsler metrics on symmetric spaces, Gromov pre-metric.
result Equality of Hausdorff dimensions and Finsler critical exponents.
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.
Study confirms optimal bounds for group cohomology of Lie groups.
problem Optimal bounds for group cohomology of Lie groups.
method Combining complementary vanishings with spectral sequences and quasi-isometry invariance.
result Non-vanishing of group Lp-cohomology for large p and equal degree to 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…
Arithmetic groups have polynomially bounded Dehn functions in certain products of Lie groups.
problem Understanding the Dehn functions of arithmetic groups in products of Lie groups.
method Utilizing results from Bestvina-Eskin-Wortman and Cornulier-Tessera.
result Arithmetic groups defined over global fields with specific properties have polynomially bounded Dehn functions.
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.
Local-to-global principle for Morse actions on symmetric spaces.
problem Recognizing Morse actions on symmetric spaces.
method Equivariant Morse quasiisometric embeddings of trees into symmetric spaces.
result Algorithmic recognizability of Morse actions and construction of Morse Schottky subgroups.
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.
We complete the quasi-isometric classification of irreducible lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero by showing that any quasi-isometry of a rank one S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic zero is…
Higher index theorem for Dirac operators on finite-volume spaces.
problem Index of Dirac operators on finite-volume spaces.
method Using algebraic K-theory and traces defined by orbital integrals.
result Nonzero and computable results for higher orbital integrals.
Study shows singularity of stationary measure on Furstenberg boundary for certain random walks.
problem Singularity of stationary measure on Furstenberg boundary for random walks.
method Analysis of random walks on semisimple Lie groups with specific properties.
result Stationary measure is singular to Lebesgue measure in certain cases.
We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly with covolume. We obtain similar results for isotropy subgroups in lattices. Ge…
Workshop notes on positivity in Lie groups and its applications.
problem Understanding total and Θ-positivity in semisimple Lie groups. method Discussion and analysis of existing theories and recent developments.
result Progress in classifying higher Teichmüller spaces through Θ-positivity. Constructs measurable equivariant maps for higher rank Lie groups.
problem Measurable cocycles with values into higher rank Lie groups.
method Extends Connell-Farb's construction to measurable cocycles.
result Constructs measurable equivariant maps with uniformly bounded Jacobian.
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be in…
Survey on discrete subgroups of symmetric spaces with rank 1 behavior.
problem Characterizing discrete subgroups with rank 1 behavior.
method Various characterizations and dynamical/geometric properties.
result Found domains of proper discontinuity and constructed compactifications.
The paper proves Zimmer's conjecture for non-uniform lattices by controlling mass escape and Lyapunov exponents.
problem Proving Zimmer's conjecture for non-uniform lattices in higher-rank semisimple Lie groups.
method Establishes finiteness of low-dimensional actions, introduces novel techniques to control mass escape and Lyapunov exponents.
result Proves Zimmer's conjecture for many non-uniform lattices, improving previous results.
Study of Anosov representations in pseudo-Riemannian hyperbolic spaces.
problem Understanding Anosov representations in higher-dimensional spaces.
method Examining representations into projective indefinite orthogonal groups and their action on H^{p,q-1}.
result Intimate connection between Anosov representations and convex cocompactness in this setting.
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.
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.
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued co…
Solves index problem for curved BGG sequences in parabolic geometry.
problem Index theory of curved Bernstein-Gelfand-Gelfand sequences.
method Utilizes K-homology and noncommutative geometry.
result Solves the index problem for BGG-sequences on flat parabolic geometry.
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.
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.
Introduces Θ-positivity in Lie groups, linking to surface group representations.
problem Understanding positivity in Lie groups and its relation to surface group representations.
method Introduces Θ-positivity as a new concept and shows its applicability to specific Lie groups. result Identifies new families of Lie groups (SO(p,q) for p<q and exceptional Lie groups) with Θ-positive structures. Let G be a higher-rank semisimple Lie group over a nonarchimedean local field, for example G=PGL(n,QP). To any lattice L in G there is an associated simplicial complex BL, given by the quotient by L of the Bruhat-Tits building associated to G. In this paper prove that the simplicial structure $B_L…
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.
Let G be a real semisimple Lie group with no compact factors and finite centre, and let Λ be a lattice in G. Suppose that there exists a homomorphism from Λ to the outer automorphism group of a right-angled Artin group AΓ with infinite image. We give an upper bound to the real rank of G that is determined by the…
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…
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…
Harmonic maps into semisimple Lie groups factored into simpler components.
problem Understanding harmonic maps into semisimple Lie groups.
method Factorization into harmonic maps with values in the components of the Iwasawa decomposition.
result Harmonic maps from \(\mathbb{R}^n\) into \(SL(2,\mathbb{R})\) studied using this factorization.
Divergence functions of a metric space estimate the length of a path connecting two points A, B at distance ≤n avoiding a large enough ball around a third point C. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. By Olshanskii-Osin-Sapir, that…