New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
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.
Proves restrictions on projective Anosov representations of hyperbolic groups.
problem Restrictions on projective Anosov representations of hyperbolic groups.
method Word hyperbolic groups and Gromov boundary analysis.
result Word hyperbolic groups with certain properties cannot admit projective Anosov representations.
Develops measures for non-Borel Anosov groups on Furstenberg boundary.
problem Measuring non-Borel Anosov groups on the Furstenberg boundary.
method Theory of Patterson--Sullivan measures, strict convexity, entropy rigidity.
result Existence, uniqueness, and ergodicity of measures on Furstenberg boundary.
Study Anosov representations of reducible suspensions of hyperbolic groups.
problem Characterize dynamical properties of reducible suspensions of Anosov representations.
method Analyzing linear representations of non-elementary hyperbolic groups, focusing on weak unipotent actions on subspaces.
result Characterize when reducible suspensions are discrete and faithful, quasi-isometrically embedded, and Anosov.
Cube complexes allow hyperbolic groups to have Anosov representations.
problem Understanding representations of hyperbolic groups in higher dimensions.
method Analyzing representations of hyperbolic groups acting on CAT(0) cube complexes.
result Generic representations of certain groups are Anosov.
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.
Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.
problem Understanding Anosov representations and their properties.
method Characterizations via equivariant limit maps, Cartan property, and uniform gap summation.
result Characterizations of Anosov representations and strongly convex cocompact subgroups.
Proves certain subgroups of genus 2 handlebody group are convex cocompact.
problem Characterizing subgroups of genus 2 handlebody group.
method Proving convex cocompactness of purely pseudo-Anosov subgroups.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact.
We characterize groups admitting Anosov representations into SL(3,R), projective Anosov representations into SL(4,R), and Borel Anosov representations into SL(4,R). More generally, we obtain bounds on the cohomological dimension of groups admitting Pk-Anosov r…
Let Γ be a one-ended, torsion-free hyperbolic group and let G be a semisimple Lie group with finite center. Using the canonical JSJ splitting due to Sela, we define amalgam Anosov representations of Γ into G and prove that they form a domain of discontinuity for the action of Out(Γ). In the appendix,…
Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.
problem Understanding the structure of pseudo-Anosov subgroups in surface bundles over tori.
method Using the Birman exact sequence to show convex cocompactness.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact in surface bundles over tori.
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.
New algorithm verifies Anosov condition for surface groups efficiently.
problem Verifying Anosov condition for surface groups in higher dimensions.
method Finite criteria to certify projective Anosov subgroups, practical algorithm.
result Practical algorithm reduces verification from 2 million to 8-length words.
We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.
The study shows pseudo-Anosovs are common in mapping class groups.
problem Counting pseudo-Anosovs in mapping class groups.
method Using weakly contracting isometries and Morse elements.
result Pseudo-Anosovs are generic in mapping class groups.
Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. 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.
Loxodromic elements are pseudo-Anosov on specific graphs.
problem Characterizing loxodromic elements in specific groups.
method Analyzing subgroups acting on multiarc and curve graphs, and the handlebody group on disk graphs.
result Loxodromic elements are pseudo-Anosov on witness graphs.
Generalizes Klein-Maskit theorem to free products of Anosov subgroups.
problem Proving a generalization of Klein-Maskit theorem for free products of Anosov subgroups.
method Analyzing Anosov subgroups in free products of semisimple Lie groups.
result The group generated by free products of Anosov subgroups is again Anosov and isomorphic to the free product.
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 SLm(K). result Higher rank Anosov representation theorems fail for m≥30. 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 Γ-conformal measures for critical dimensions. 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.
Characterizes Anosov reducible representations in terms of eigenvalues.
problem Understanding Anosov representations in reducible settings.
method Characterizes Anosov representations using eigenvalue magnitudes of irreducible block factors.
result Connected components of character varieties do not contain reducible representations for many non-elementary hyperbolic groups.
The paper proves rigidity results for Anosov flows and their orbit equivalences.
problem Characterizing orbit equivalences of Anosov flows and their dynamics.
method Using hyperbolic-like dynamics, the paper proves a spectral rigidity theorem and gives efficient criteria for orbit equivalences.
result Characterizes orbit equivalent flows in terms of fundamental group elements represented by periodic orbits.
New findings on generating mapping class groups using pseudo-Anosov elements.
problem Generating mapping class groups using specific types of elements.
method Proving the generation of mapping class groups by pseudo-Anosov elements and conjugate reducible but not periodic elements.
result The mapping class group can be generated by two conjugate pseudo-Anosov elements with arbitrarily large dilatations for surfaces of genus greater than or equal to nine.
Positive representations of surface groups in PO(p,q) form connected components of character varieties.
problem Characterizing representations of surface groups in special orthogonal groups PO(p,q).
method Using Anosov representations and root versus weight collar lemmas.
result Connected components of character varieties are formed by Θ-positive Anosov representations. We define the notion of affine Anosov representations of word hyperbolic groups into the affine group SO0(n+1,n)⋉R2n+1. We then show that a representation ρ of a word hyperbolic group is affine Anosov if and only if its linear part Lρ is Anosov in SO0(n+1,n) with …
The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.
problem Characterizing Anosov subgroups of Sp(2n,R) based on subset Θ.
method Analyzing the structure of Anosov subgroups in terms of subset Θ.
result Anosov subgroups of Sp(2n,R) are virtually free or surface groups if Θ contains an odd integer, otherwise they are not.
Anosov representations of surface groups are complex manifolds.
problem Characterizing the geometry of Anosov representations of surface groups.
method Analyzing the character varieties of Anosov representations into $\SL(n , \C)$.
result Character varieties of Anosov representations are complex manifolds of specific dimension.
Exponential proportion of pseudo-Anosovs in mapping class groups.
problem Proportion of non-pseudo-Anosov mapping classes in a ball of radius R.
method Using word metric and finite generating sets, we show that the proportion decreases exponentially.
result Proportion of non-pseudo-Anosov mapping classes decreases exponentially.
New representations defined for groups and graphs, with applications to stable representations.
problem Defining and constructing new types of representations for groups and graphs.
method Introducing (R,Λ)-directed Anosov representations and using Fock-Goncharov positivity to construct them. result Constructs large families of primitive stable representations from F2 to PGL(V), including non-discrete and non-faithful examples. Anosov deformations created for surfaces with specific properties.
problem Creating proper slices in character varieties for specific surface groups.
method Constructing proper slices in character varieties for surface groups into SL(3,R).
result Representations are Borel Anosov.
Theory of relatively Anosov representations using flow methods.
problem Developing a theory for relatively Anosov representations.
method Using the contracting flow on a bundle to define and study relatively Anosov representations.
result Definition and study of uniformly relatively Anosov representations and a stability result.
The paper introduces cataclysm deformations for Anosov representations.
problem Deforming Anosov representations in Lie groups.
method Constructing cataclysm deformations for θ-Anosov representations into semisimple Lie groups. result Cataclysm deformations are injective for Hitchin representations but not for all θ-Anosov representations. Proves nonemptyness of domains for specific group actions.
problem Nonemptyness of domains of proper discontinuity for Anosov groups of affine Lorentzian transformations.
method Proof of nonemptyness of domains of proper discontinuity.
result Proves nonemptyness of domains for Anosov groups of affine Lorentzian transformations.
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.
In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any …
New Klein-Maskit theorems for Anosov subgroups.
problem Creating new Kleinian groups from old ones.
method Analogous combination theorems for Anosov subgroups.
result Construct new Kleinian groups using old ones for Anosov subgroups.
This thesis explores bifoliated planes and their group actions from Anosov flows.
problem Understanding bifoliated planes and their group actions from Anosov flows.
method Analyzes bifoliated planes associated with Anosov flows and describes their properties.
result Shows left-orderability of groups acting on bifoliated planes and describes bifoliated planes from specific families of Anosov flows.
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 subgroups generalize convex-cocompact groups in hyperbolic geometry.
problem Understanding convex-cocompact subgroups in higher rank geometry.
method Characterizing Anosov subgroups and comparing them to convex-cocompact groups.
result Anosov subgroups are the right generalizations of convex-cocompact groups in hyperbolic geometry.
New subgroup behavior in genus-2 mapping class group identified.
problem Understanding subgroups in genus-2 mapping class group.
method Analyzing purely pseudo-Anosov subgroups as convex cocompact.
result Finitely-generated, purely pseudo-Anosov subgroups are convex cocompact.
The paper constructs a free abelian group from Anosov representations on bundles.
problem Constructing a free abelian group from Anosov representations on bundles.
method Extending the Γ-action via ρ to the space of connections on pullbacks of tangent bundles. result A free abelian group Fab is constructed and acts properly discontinuously on it. We define new topological invariants for Anosov representations and study them in detail for maximal representations of the fundamental group of a closed oriented surface into the symplectic group.
Study shows certain subgroups of fibered 3-manifolds are convex cocompact.
problem Understanding subgroups of fibered 3-manifolds in mapping class groups.
method Used the Birman exact sequence to show convex cocompactness.
result Finitely generated pseudo-Anosov subgroups are convex cocompact.