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.
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 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.
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.
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…
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…
Anosov subgroups' deformations affect limit cones and growth indicators continuously.
problem Understanding continuous changes in Anosov subgroups' effects on limit cones and growth indicators.
method Continuous variation of limit cones and growth indicators under deformations of Anosov subgroups, with convexity assumptions.
result Limit cones and growth indicators vary continuously under deformations of Anosov subgroups.
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.
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.
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.
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.
The paper defines Dirichlet domains for Anosov subgroups in Lie groups.
problem Finding finite-sided Dirichlet domains for Anosov subgroups in Lie groups.
method Introducing a sufficient condition for finite-sided Dirichlet domains in semisimple Lie groups for polyhedral Finsler metrics.
result Finitely generated subgroups of semisimple Lie groups can have finite-sided Dirichlet domains under certain conditions.
Anosov subgroup equidistributes geodesics and holonomies on homogeneous spaces.
problem Equidistribution of geodesics and holonomies in Anosov homogeneous spaces.
method Analyzes maximal flat cylinders and their holonomies for Anosov subgroups.
result Joint equidistribution of maximal flat cylinders and holonomies as circumference tends to infinity.
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.
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.
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…
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.
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.
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.
The paper proves a unique conformal measure for Anosov groups and shows local mixing.
problem Proving the uniqueness of conformal measures for Anosov groups.
method Analogue of Sullivan's theorem for Anosov subgroups of semisimple groups.
result Uniqueness of conformal measures and local mixing for 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.
Given any generating set of any pseudo-Anosov-containing subgroup of the mapping class group of a surface, we construct a pseudo-Anosov with word length bounded by a constant depending only on the surface. More generally, in any subgroup G we find an element f with the property that the minimal subsurface supporting a …
Study of pseudo-Anosov actions on SU(2)-character variety for genus 2 surfaces.
problem Characterizing pseudo-Anosov actions on SU(2)-character variety. method Analysis of mapping class group subgroups and their pseudo-Anosov elements.
result Existence of a subgroup containing infinitely many pseudo-Anosov elements with invariant rational functions.
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. 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.
Constructs flows on quotients of Lie groups for Anosov subgroups.
problem Understanding dynamics on quotients of Lie groups by Anosov subgroups.
method Utilizes geometric structures and Lie group theory to construct and analyze flows.
result Establishes that all refraction flows arise from this construction.
Study counts and parametrizes flag components in SO0(p,q) space.
problem Counting and characterizing flag components in SO0(p,q) space.
method Parametrization and computation of Plücker coordinates.
result Anosov subgroups are virtually isomorphic to surface or free 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.
Anosov groups in rank ≤3 have unique ergodic horospherical actions.
problem Characterizing unique ergodicity of horospherical actions for Anosov groups.
method Analyzing N-action on Rv for v in the limit cone. result Unique ergodicity holds for r≤3 and v in the limit cone. The paper describes decompositions of geometric measures on Anosov homogeneous spaces.
problem Decomposing geometric measures on Anosov homogeneous spaces.
method Ergodic decompositions of Burger-Roblin and Bowen-Margulis-Sullivan measures.
result The space of non-trivial invariant ergodic measures is homeomorphic to a product space.
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 paper studies invariant measures for specific actions in algebraic groups.
problem Investigating invariant measures for horospherical actions and Anosov groups.
method Analyzing the space of invariant measures for NM-actions on Γ\G. result The space of invariant measures is homeomorphic to RextrankG−1. Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.
problem Investigating the Hausdorff dimension of Anosov subgroup limit sets with self-affine complexity.
method Analyzing the Hausdorff dimension of projective limit sets Λ1(Γ) of Anosov subgroups Γ under specific assumptions about their affine complexity. result The Hausdorff dimension of Λ1(Γ) is determined by the critical exponent of the first simple root under partial quasi-self-similarity. Regular subgroups of SL3(R) are identified and ruled out.
problem Identifying and characterizing regular subgroups of SL3(R).
method Using Kapovich–Leeb–Porti and Guichard–Wienhard divergent subgroups criteria, and Oh's results.
result Regular subgroups of SL3(R) are precisely lattices in minimal horospherical subgroups.
Anosov flows found on many hyperbolic 3-manifolds.
problem Existence of Anosov flows on hyperbolic 3-manifolds.
method Explicit construction of Anosov flows on fibered hyperbolic 3-manifolds.
result Positive density of fibered hyperbolic manifolds carrying Anosov flows.
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.
Paper proves non-arithmetic Teichmüller length spectra for subgroup of mapping class groups.
problem Proving non-arithmetic Teichmüller length spectra for subgroups of mapping class groups.
method Introducing cross-ratios on Teichmüller and projectable mapping classes, studying their geometric and dynamical properties.
result Every non-elementary subgroup of the mapping class group has non-arithmetic Teichmüller length spectrum.
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.
We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.
The study examines translation lengths of pseudo-Anosov maps on curve graphs.
problem Estimating translation lengths of pseudo-Anosov maps on curve graphs.
method Analyzing geodesic axes and powers of Dehn twists.
result Determining minimal translation lengths and optimizing map ratios.
Study proves projective Anosov subgroups lead to mixing flows in specific spaces.
problem Understanding mixing properties of flows on specific geometric spaces.
method Constructing non-empty domain of discontinuity in homogeneous space, using spectral estimates for transfer operators.
result Exponential mixing, spectral gap, and meromorphic continuation of zeta functions established.
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.
Anosov groups study matrix coefficients and orbit counting in symmetric spaces.
problem Anosov groups and their matrix coefficients in symmetric spaces.
method Asymptotic analysis of matrix coefficients and higher rank measures.
result Asymptotic behavior of matrix coefficients and orbit counting results.
Combination theorems for convex projective geometry subgroups.
problem Understanding discrete subgroups in convex projective geometry.
method General combination theorems for discrete subgroups preserving properly convex open subsets.
result Free products of convex cocompact subgroups are convex cocompact.
New domains of discontinuity found for Anosov representations.
problem Understanding Anosov representations acting on homogeneous spaces.
method Constructing open domains of discontinuity for Anosov representations acting on specific homogeneous spaces.
result Describes the largest possible open domains of discontinuity for Zariski dense Anosov representations.
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.
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.
We prove that a word hyperbolic group which admits a P2q+1-Anosov representation into PGL(4q+2,R) contains a finite-index subgroup which is either free or a surface group. As a consequence, we give an affirmative answer to Sambarino's question for Borel Anosov representations into $\mathsf{SL}…