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.
Any closed, oriented, hyperbolic three-manifold with nontrivial second homology has many quasigeodesic flows, where quasigeodesic means that flow lines are uniformly efficient in measuring distance in relative homotopy classes. The flows are pseudo-Anosov flows which are almost transverse to finite depth foliations in …
Develops theory of relatively Anosov representations using flow examples.
problem Understanding relatively Anosov representations.
method Uses a contracting flow on a bundle to define Anosov representations and builds examples.
result Builds families of examples of relatively Anosov representations.
Anosov geodesic flow proven in non-compact manifolds with negative curvature.
problem Proving Anosov geodesic flow in non-compact manifolds with negative curvature.
method Proving the geodesic flow is Anosov by showing average sectional curvature is negative and uniformly away from zero.
result Constructed a non-compact manifold with Anosov geodesic flow.
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.
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.
Study shows exact dimensionality and regularity of manifolds for specific groups.
problem Exact dimensionality and regularity of manifolds for relatively Anosov groups.
method Dynamical methods, including finite and mixing of Bowen–Margulis–Sullivan measures.
result Manifolds are C1-regular and growth indicator is strictly concave. 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.
We consider a billiard in the sphere S^2 with circular obstacles, and give a sufficient condition for its flow to be uniformly hyperbolic. We show that the billiard flow in this case is approximated by an Anosov geodesic flow on a surface in the ambiant space S^3. As an application, we show that every orientable surfac…
We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a …
Study on measurable pseudo-Anosov maps on surfaces.
problem Characterize dynamics of pseudo-Anosov maps on surfaces.
method Analyze measurable pseudo-Anosov homeomorphisms with specific properties.
result Prove transitivity, dense periodic points, sensitivity, and ergodicity.
We consider a totally nonsymplectic Anosov action of Z^k which is either uniformly quasiconformal or pinched on each coarse Lyapunov distribution. We show that such an action on a torus is C^\infty--conjugate to an action by affine automorphisms. We also obtain similar global rigidity results for actions on an arbitrar…
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.
Cataclysm deformations study Anosov representations and their convergence.
problem Understanding convergence of Anosov representations under deformation.
method Cataclysm deformation of Anosov representations using twisted transverse cocycles.
result Uniform convergence of cataclysm deformations on compact sets.
New insights into Anosov representations of hyperbolic groups.
problem Understanding Anosov representations of relatively hyperbolic groups.
method Proving representations can be interpreted as restricted Anosov representations over flow spaces and showing stability under deformations.
result Representations of certain types are divergent, extended geometrically finite and stable under small deformations.
Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.
problem Understanding Anosov representations and their deformations.
method Cataclysm deformations based on twisted transverse cocycles.
result Uniform convergence of cataclysm deformations on compact sets.
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.
We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surfa…
The paper studies stability of discretized Anosov flows.
problem Global stability of discretized Anosov flows.
method Defined and proved equivalence with previous definitions, showed properties through C1 openness and closedness, and established integrability and uniqueness of invariant foliations. result Discretized Anosov flows are globally stable.
Any smooth surface in R^3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R^2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive and has finite horizon, then …
Study on 3-manifolds admitting pseudo-Anosov maps on subsurfaces.
problem Which 3-manifolds admit pseudo-Anosov maps on incompressible subsurfaces?
method Determine self-homeomorphisms of 3-manifolds that restrict to pseudo-Anosov maps on subsurfaces.
result Self-homeomorphisms of irreducible 3-manifolds are isotopic to partially pseudo-Anosov homeomorphisms.
Study equidistribution for flows on geometrically finite convergence group actions.
problem Counting, mixing and equidistribution for flows on geometrically finite convergence group actions.
method Establishing results for finite BMS measures on flow spaces associated to geometrically finite convergence group actions.
result Results apply to flow spaces associated to relatively Anosov groups.
Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.
problem Characterizing Borel Anosov subgroups in SL(d,R).
method Analysis of antipodal subsets and quasi-isometric embeddings.
result Borel Anosov subgroups are virtually free or hyperbolic surface groups.
We show that a self orbit equivalence of a transitive Anosov flow on a 3-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is R-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption…
We propose several common extensions of the classes of Anosov subgroups and geometrically finite Kleinian groups among discrete subgroups of semisimple Lie groups. We relativize various dynamical and coarse geometric characterizations of Anosov subgroups given in our earlier work, extending the class from intrinsically…
Formula connects length and correlation functions via ghost polygons and Poisson bracket.
problem Understanding functions on moduli spaces of Anosov representations.
method Introduced ghost polygons and ghost algebra to compute Poisson bracket.
result Stability of length and correlation functions under Poisson bracket.
New family of measurable pseudo-Anosov maps on spheres.
problem Generalizing pseudo-Anosov maps to measurable ones.
method Continuous family of homeomorphisms on sphere, semi-conjugate to core tent map.
result Measurable pseudo-Anosov maps have invariant dense streamlines with uniform measures.
Characterizes transverse surfaces for pseudo-Anosov flows in 3-manifolds.
problem Characterizing surfaces transverse to pseudo-Anosov flows.
method Correspondence between surfaces and veering triangulations, Thurston norm minimization.
result Thurston-norm minimizing surfaces are almost transverse to pseudo-Anosov flows.
Strict concavity proven for growth indicator function of certain groups.
problem Proving strict concavity of growth indicator function for specific groups.
method Smoothness of Manhattan hypersurface and critical-exponent map.
result Strict concavity of growth indicator function for relatively Anosov groups.
Simplified approach to pseudo-Anosov flows on 3-manifolds.
problem Complexity in understanding pseudo-Anosov flows on 3-manifolds.
method Streamlined framework called Anosov-like group actions.
result Unified and simplified presentation of pseudo-Anosov flows.
Uniformly branching trees are equivalent to certain metric spaces.
problem Characterizing metric spaces equivalent to uniformly branching trees.
method Proving equivalence between trivalent quasiconformal trees and uniformly branching trees.
result Any two uniformly branching trees are quasisymmetrically equivalent.
New measure of maximal entropy found for a class of geometrically finite groups.
problem Finding a measure of maximal entropy for relatively Anosov groups.
method Constructing reparameterizations and using exponential expansion along unstable foliations.
result The Bowen-Margulis-Sullivan measure is finite and unique for relatively Anosov groups.
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.
We show that for any group G that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then G acts properly on a uniformly convex Banach space as well.
New stability estimate for metric rigidity in hyperbolic dynamics.
problem Metric rigidity in hyperbolic dynamics.
method Radial source estimates in Hölder-Zygmund spaces for uniformly hyperbolic dynamics.
result Metrics with same marked length spectrum are isometric in C3+ε-close metrics in any dimension ≥2. Let Γ be a finitely generated group which is hyperbolic relative to a finite family {H1,...,Hn} of subgroups. We prove that Γ is uniformly embeddable in a Hilbert space if and only if each subgroup Hi is uniformly embeddable in a Hilbert space.
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.
Uniform K-homology theory applied to elliptic operators on manifolds with boundary.
problem Developing a theory to study boundary conditions for elliptic operators on non-compact manifolds.
method Theory of relative uniform K-homology, developing a relative index map.
result Uniform K-homology classes of boundary conditions and their connection to the higher ρ-invariant.
Study eta invariant on non-compact manifolds with positive scalar curvature.
problem Proving geometric formulas and index theorems for uniformly positive scalar curvature metrics.
method Using Dirac-Schrödinger operators and relative eta invariant.
result New geometric formula for spectral flow and index formula for uniformly positive scalar curvature metrics.
Extremal metrics exist if uniformly K-stable over models.
problem Existence of extremal metrics on complex projective varieties.
method Uniform K-stability over models of extremal tori. result Extremal metrics exist if uniformly K-stable. The study proves conditions for hyperbolic isometries on fine curve graphs of higher genus surfaces.
problem Conditions for hyperbolic isometries on fine curve graphs of higher genus surfaces.
method Proves equivalence of conditions involving isotopic maps, pseudo-Anosov maps, and ergodic rotation sets.
result Ergodic homological rotation sets have nonempty interior for certain isotopic maps.
We give a geometric interpretation of the maximal Satake compactification of symmetric spaces X=G/K of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable G-invariant Finsler metric on X. As an application, we establish the existence of natural bordifications, as orbifold…
Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian manifold with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. …
The study examines growth of quadratic forms under Anosov subgroups.
problem Growth of quadratic forms under Anosov subgroups.
method Analyzes exponential bounds and asymptotic counting functions for distances between geodesic copies of symmetric spaces.
result Shows asymptotic behavior of counting functions for certain choices of quadratic forms.
Entropy rigidity theorem for cusped Hitchin representations.
problem Entropy rigidity for Hitchin representations of cusped groups.
method Introduction of (1,1,2)-hypertransverse groups and transverse representations.
result Hausdorff dimension of conical limit set agrees with simple root entropy.
In this paper, we study the Hausdorff dimension of the Floyd and Bowditch boundaries of a relatively hyperbolic group, and show that for the Floyd metric and shortcut metrics respectively, they are are both equal to a constant times the growth rate of the group. In the proof, we study a special class of conical points …
Let δg,n be the minimal dilatation of pseudo-Anosovs defined on an orientable surface of genus g with n punctures. Tsai proved that for any fixed g≥2, the logarithm of the minimal dilatation logδg,n is on the order of nlogn. The main result of this paper is that if 2g+1 is relativel…
We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furt…