The paper justifies two questions on special maps on subgroup of reals.
problem Justifying two questions on special maps on subgroup of reals.
method Different points of view and discussion of two versions of Anderson's Involution Conjecture.
result Discussion of two versions of Anderson's Involution Conjecture.
The study examines discrete subgroups of Lie groups and their residual finiteness.
problem Determining when discrete subgroups of Lie groups are residually finite.
method Analyzes known results and poses open questions.
result Answers to open questions will provide a comprehensive understanding of residual finiteness in Lie groups.
Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.
problem Characterizing dense subgroups of algebraic groups.
method Bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups.
result No C1-smooth slim limit set for higher rank semisimple algebraic groups. The study finds discrete subgroups with full limit sets in higher rank Lie groups.
problem Finding discrete subgroups with full limit sets in higher rank Lie groups.
method Analyzing real semi-simple Lie groups of higher rank and providing criteria for discrete subgroups of G=SL(3,R). result Existence of discrete subgroups with full limit sets in higher rank Lie groups.
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 describes orbits of parabolic subgroups in complexified actions.
problem Understanding orbits of parabolic subgroups in complexified actions.
method Analyzes the gradient map μₚ to describe orbits of parabolic subgroups.
result Describes compact orbits of parabolic subgroups in terms of the gradient map.
Automates subgroup discovery for real-valued targets using prior knowledge.
problem Finding meaningful patterns in high-dimensional, real-valued data.
method Subjective Interestingness framework FORSIED for efficient subgroup discovery.
result Automatically discovers informative subgroups in data for real-valued targets.
The study shows how discrete subgroups' critical exponents relate to their Zariski density in certain groups.
problem Understanding the density of discrete subgroups in semisimple Lie groups.
method Critical exponents and unitary representations.
result Discrete subgroups with critical exponents greater than a certain value are Zariski dense.
We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation of real trees. Our main result states that a finitely generated group G acts n…
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.
FairVis helps discover biases in machine learning models.
problem Discovering biases in machine learning models is challenging due to multiple definitions of fairness and numerous subgroups.
method Integrates a novel subgroup discovery technique with a mixed-initiative visual analytics system.
result Demonstrates how FairVis helps discover biases in real datasets.
Geodesic orbit metrics on real flag manifolds identified.
problem Classifying real flag manifolds with geodesic orbit metrics.
method Investigated invariant metrics on real flag manifolds, focusing on those where geodesics are orbits of one-parameter subgroups.
result Non-trivial geodesic orbit metrics exist on real flag manifolds, unlike in the complex case.
We show that if Γ is an irreducible subgroup of SU(2,1), then Γ contains a loxodromic element A. If A has eigenvalues λ1=λeiφ, λ2=e−2iφ, λ3=λ−1eiφ, we prove that Γ is conjugate in SU(2,1) to a subgroup of SU(2,1,Q(Γ,λ)), where $\mat…
Let G be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of G, which can be also seen as the space of maximal flats of the symmetric space of G. We define its Chabauty compactification…
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.
We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theor…
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.
Criterion for congruence RFRS towers in hyperbolic lattices.
problem Criteria for hyperbolic lattices to have congruence RFRS towers.
method Criterion based on congruence subgroups.
result Virtually fibered Bianchi groups and new RFRS Kähler groups.
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.
We partially describe equivariant Dirac and generalized complex structures on a homogeneous space G/K by giving equivalent data involving only the Lie algebra. We consider real semisimple adjoint orbits in any semisimple Lie algebra over R and real nilpotent orbits in sln(R). We give a complete …
Empirical study on rich subgroup fairness for machine learning.
problem Ensuring fairness across large subgroups in machine learning.
method An algorithm that learns subject to rich subgroup fairness constraints.
result Rich subgroup fairness leads to large gains in fairness with mild accuracy costs.
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. 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.
Model patching closes subgroup performance gaps in skin cancer classification.
problem Inconsistent model performance on specific subgroups of a class.
method Two-stage framework that models subgroup features and learns semantic transformations, followed by data augmentation.
result Reductions in robust error of up to 33% relative to best baseline on benchmark datasets.
The paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.
problem Defining and characterizing parabolic subgroups in complex braid groups.
method Introducing and studying parabolic subgroups of generalized braid groups associated with complex reflection groups.
result Parabolic subgroups form a lattice in most cases, with specific properties and conjectures about hyperbolicity.
Study convex cocompact subgroups in real projective geometry.
problem Characterize discrete subgroups acting on real projective space.
method Define and characterize convex cocompactness, extend results from orthogonal groups.
result Equivalence of different convex cocompactness conditions for word hyperbolic groups.
A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.
Chiseling finds valid subgroups interactively, improving on existing methods.
problem Finding valid subgroups with inferential guarantees in regression and causal inference.
method Interactive subgroup refinement with inferential validity guarantees.
result Chiseling identifies better subgroups than existing methods with inferential guarantees.
New algorithm tackles subgroup fairness in AI with multiple sensitive attributes.
problem Heavy computational burdens and data sparsity in subgroup fairness for multiple sensitive attributes.
method Doubly Regressing Adversarial learning (DRAF) for subgroup fairness, focusing on subgroups with sufficient sample sizes and marginal fairness.
result DRAF algorithm reduces a surrogate fairness gap for supIPM with less computation than directly reducing supIPM.
New method identifies subgroups in censored data.
problem Identifying meaningful patterns in heterogeneous populations.
method Combining inverse probability weighting, M-estimation, and concave pairwise fusion penalization.
result Robust approach for censored data under heterogeneous AFT models.
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.
Improves fairness in machine learning by adding underrepresented group data.
problem Machine learning biases across subgroups due to under-representation or societal biases.
method Data augmentation via pairwise mixup across subgroups to balance subpopulations.
result Achieves fair outcomes with robust if not improved accuracy.
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.
Bayesian Supervised Causal Clustering identifies patient subgroups for personalized decision-making.
problem Finding patient subgroups with similar characteristics for personalized decision-making.
method Bayesian Supervised Causal Clustering (BSCC) that identifies homogenous subgroups based on treatment effects.
result BSCC identifies subgroups with similar covariate profiles and treatment effects.
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.
Study of parabolic vector bundles on Klein surfaces.
problem Understanding parabolic vector bundles on Klein surfaces.
method Defined and studied parabolic vector bundles, proving isomorphism classes correspond to representations of discrete subgroups.
result Isomorphism classes of polystable real and quaternionic parabolic vector bundles correspond to equivalence classes of representations of discrete subgroups.
Proposes a method to learn fair predictors for multiple subgroups with limited data.
problem Fairness and accuracy issues in learning from multiple subgroups with limited data.
method Formulates a bilevel objective to learn subgroup-specific predictors and a fair predictor that is close to all of them.
result The method effectively controls group sufficiency and generalization error, improving fairness and accuracy.
SHIFT framework identifies subgroups with large ML model performance decay.
problem Large model performance decay in subgroups when deployed.
method Subgroup-scanning Hierarchical Inference Framework (SHIFT) for performance drift.
result SHIFT identifies interpretable subgroups with large performance decay and suggests targeted actions to mitigate it.
The paper finds dense subgroups in certain Lie groups.
problem Finding dense subgroups in Lie groups.
method Constructing dense surface subgroups in specific Lie groups.
result Uniform lattices contain infinitely many dense Hitchin representations.
Study on G2-type flag manifolds, focusing on invariant metrics and Ricci flow.
problem Characterizing and analyzing metrics on G2-type flag manifolds. method Investigation of invariant metrics, analysis of g.o. metrics, and Ricci flow techniques.
result Characterization of metrics invariant under maximal compact subgroups.
Two groups with specific limit sets in hyperbolic spaces are identified.
problem Identifying convex cocompact subgroups with specific limit sets in real hyperbolic spaces.
method Examples of subgroups generated by reflections and rotations with limit sets as Pontryagin spheres and Menger curves.
result Examples of convex cocompact subgroups with limit sets as Pontryagin spheres and Menger curves are found.
In subgroup discovery, also known as supervised pattern mining, discovering high quality one-dimensional subgroups and refinements of these is a crucial task. For nominal attributes, this is relatively straightforward, as we can consider individual attribute values as binary features. For numerical attributes, the task…
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
The center of the group of quasi-isometries of the real line is trivial.
problem Identifying the center of the group of quasi-isometries of the real line.
method Analyzing the group structure and using the quasi-isometry properties to show the triviality of the center.
result The center of the group of quasi-isometries of the real line is trivial.
The paper studies cohomologies of hypercomplex manifolds and their dimensions.
problem Understanding cohomologies and dimensions of invariant and anti-invariant subgroups.
method Proving a compact hypercomplex manifold is C∞-pure-and-full under certain conditions and studying dimensions of subgroups. result Characterization of hyperkähler with torsion metrics in terms of the dimension of the Jˉ-invariant subgroup. Study critical exponents in normal subgroups of higher rank Lie groups.
problem Understanding critical exponents in normal subgroups of higher rank Lie groups.
method Analyzing subgroups and their critical exponents in a higher rank semi-simple Lie group.
result Critical exponents of normal subgroups coincide under certain conditions.
Proposes a method to identify subgroup structure and estimate covariate effects for multivariate response data.
problem Identifying subgroup structure and estimating covariate effects in multivariate response data.
method Joint heterogeneity and reduced-rank learning framework using rank-constrained pairwise fusion penalization.
result Established the asymptotic properties of the estimators and proposed a predictive information criterion for rank selection.