Paper proves almost all stabilizer subgroups of Thompson's group satisfy Alexander's theorem.
problem Alexander's theorem for stabilizer subgroups of Thompson's group.
method Defined a method to construct knots and links from Thompson's group F and proved Alexander's theorem for stabilizer subgroups.
result Almost all stabilizer subgroups under the natural action on the unit interval satisfy Alexander's theorem.
The paper uses permutation representations to visualize group extensions and subgroups.
problem Visualizing and understanding group extensions and subgroups.
method Developing metaphoric rope-thread diagrams to represent semi-direct products and their constituents.
result Injective homomorphisms into semi-direct products are established.
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.
Recent advances in discrete subgroups of SL2(C) influenced by Thurston's work.
problem The dynamics of discrete subgroups of SL2(C)
method Discussion of recent advances in the topic
result Influenced by Thurston's work, recent advances have been made in the dynamics of discrete subgroups of SL2(C)
This paper provides an explicit form for symmetric differentials and their corresponding holomorphic functions.
problem Understanding the correspondence between symmetric differentials and L2 holomorphic functions on quotient spaces. method Explicit description of the correspondence between symmetric differentials and weighted L2-holomorphic functions. result Derivation of several applications based on the explicit form of the correspondence.
Flat affine subvarieties found in OT-manifolds.
problem Characterizing subvarieties in Oeljeklaus-Toma manifolds.
method Analyzing the structure of Oeljeklaus-Toma manifolds using number-theoretic data.
result Any complex subvariety of smallest possible positive dimension in an OT-manifold is flat affine.
We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive b2 obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions of free groups. A special case is the HNN extension associated to the endomorphi…
Study on geodesics and dihedral groups in lattices.
problem Growth and distribution of conjugacy classes of dihedral subgroups.
method Generalizing earlier work on reciprocal geodesics, proving equidistribution.
result Reciprocal geodesics are equidistributed in the unit tangent bundle.
Optimization problem on sphere identifies Minkowski functional for non-transitive groups.
problem Optimizing Minkowski functional for non-transitive subgroups of orthogonal group.
method Analytic tools and optimization on sphere to characterize subgroups.
result Characterization of non-transitive subgroups and their ranks.
The study identifies patient subgroups with enhanced or diminished opioid treatment effects.
problem Lack of prescribing guidelines for opioids leading to adverse outcomes.
method Generative model using mixture distribution and sparsity to discover subgroups with treatment effects.
result Human-interpretable insights on subgroups with enhanced or diminished treatment effects.
The article studies a monoid of smooth maps on Lie groupoids and their properties.
problem Investigating a monoid of smooth mappings on Lie groupoids and their properties.
method Generalization of A. Stacey's result (Stacey-Roberts Lemma) to infinite-dimensional manifolds.
result The group of units of a Lie groupoid is an infinite-dimensional Lie group and a Lie subgroup of the monoid.
In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f…
A left-invariant sub-Riemannian metric d on the shortened Lorentz group SO0(2,1) under the condition that d is right-invariant relative to the orthogonal Lie subgroup 1⊗SO(2) is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup 1⊗SO(2) with the an…
We consider discrete subgroups Gamma of the simply connected Lie group SU~(1,1), the universal cover of SU(1,1), of finite level, i.e. the subgroup intersects the centre of SU~(1,1) in a subgroup of finite index, this index is called the level of the group. The Killing form induces a Lorentzian metric of constant curva…
In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form H5/Γ where Γ is a torsion-free subgroup of minimal index of the congruence two subgroup Γ25 of the group Γ5 of positive units of the Lorentzian quadratic form x12+...+x52−x62. We also show that…
The purpose of this paper is to give presentations for projective S-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field Q. To our knowledge, this provides the first explicit presentations of an S-arithmetic lattice in a semisimple Lie group with S large. In particular, we…
Given a matrix A∈SL(N,Z), form the semidirect product G=ZN⋊AZ where the Z factor acts on ZN by A. Such a G arises naturally as the fundamental group of an N-dimensional torus bundle which fibers over the circle. In this paper we prove that if A has distinct eigenvalues not lying on the…
Method provides statistical guarantees for identifying subgroups in ML studies.
problem Bias and noise in estimating conditional average treatment effects (CATE).
method Develops uniform confidence bands (GATES) for estimating group average treatment effects (GATEs).
result Identifies subgroups with statistical guarantees, regardless of effect size.
Proves uniform index bound for loops in Alexandrov spaces.
problem Uniform index bound for loops in Alexandrov spaces.
method Based on ideas from Kapovitch, Petrunin, and Tuschmann; uses Hurewicz fibration and gradient push.
result Uniform index bound of w(n) for loops in Alexandrov spaces. The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…
Motivated by the celebrated Schoen-Yau-Gromov-Lawson surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold carries a metric of positive scalar curvature and an isoparametric foliation …
Study Nijenhuis operators on homogeneous spaces related to C*-algebras.
problem Characterize Nijenhuis operators on homogeneous spaces of C*-algebras.
method Analyze vector bundle maps induced by admissible operators on C*-algebras.
result Identify conditions for vector bundle maps to be Nijenhuis operators.
Framework for discovering treatment benefits in user segments.
problem Discovering differential impacts of treatments across user subgroups.
method Combines causal inference and machine learning for user segment discovery.
result Unified approach for treatment benefit discovery and assignment.
Groups with certain properties have invariant subalgebra rigidity.
problem Invariant subalgebra rigidity in groups with specific properties.
method Analyzing normal subgroups and invariant subalgebras in groups.
result Torsion-free acylindrically hyperbolic groups and hyperbolic groups have the relative ISR property.
Space of hyperbolic surfaces is path-connected.
problem Topology of hyperbolic surfaces and their subspaces.
method Constructing paths using Fenchel-Nielsen coordinates and shrinking curves.
result Path-connectivity of the space of hyperbolic surfaces.
The study uses transfer learning to compare surgical outcomes across racial/ethnic subgroups.
problem Difficulty in comparing surgical outcomes due to racial/ethnic and geographic differences.
method Causal inference framework and transfer learning to incorporate data from multiple populations.
result Racial and ethnic differences in surgical outcomes are found, with non-Hispanic Black patients experiencing wide variability.
Model predicts COVID-19 progression with interpretability.
problem Accurate and credible forecasting of COVID-19 progression.
method Integrates machine learning into disease modeling, uses interpretable encoders.
result More accurate forecasts than state-of-the-art alternatives.
We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K…
Given for instance a finite volume negatively curved Riemannian manifold M, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of M and their linear divergence rates under the geodesic flow. As…
The paper confirms conjectures about Stein manifolds formed by quotients of the ball.
problem Characterizing Stein manifolds formed by quotients of the ball.
method Analyzing discrete subgroups of PU(n,1) and their quotients.
result The quotient of the ball by geometrically finite groups is Stein.
S. Alesker has shown that if G is a compact subgroup of O(n) acting transitively on the unit sphere Sn−1 then the vector space ValG of continuous, translation-invariant, G-invariant convex valuations on Rn has the structure of a finite dimensional graded algebra over R satisfying Poincare duality. We s…
New metrics for information geometry and machine learning from Lie groups.
problem Traditional mean methods in data science and machine learning.
method Cartan-Schouten metrics on Lie groups.
result Cartan-Schouten metrics offer advantages over traditional means.
Solves Minkowski problem for affine invariant convex domains.
problem Finding convex sets with given area measures in affine spaces.
method Variational method using Steiner formula and covolume functional.
result Solves the affine invariant Minkowski problem.
Study polynomial trace identities in $SL(2,\IC)$ using quaternion algebras.
problem Understanding polynomial trace identities in $SL(2,\IC)$ and their applications.
method Use quaternion algebras over indefinites and their units to study discrete subgroups of $SL(2,\IC)$.
result Obtained structure theorems for quaternion algebras and new polynomial trace identities.
The paper calculates subgroup distortions in 3-manifold groups.
problem Understanding subgroup distortions in 3-manifold groups.
method Computed all finitely generated subgroups of finitely generated 3-manifold groups and analyzed their distortions.
result Subgroup distortions in 3-manifold groups are linear, quadratic, exponential, or double exponential.
We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem: Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on…
Stable and Morse subgroups coincide in mapping class groups.
problem Understanding subgroup properties in mapping class groups.
method Analyzing stability and Morse properties in mapping class groups.
result Stability and Morse properties coincide for subgroups of infinite index in mapping class groups.
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.
Study shows lower central subgroups of a subgroup don't contain those of the free group.
problem Whether lower central subgroups of a subgroup contain those of the free group.
method Analyzes the relationship between lower central subgroups of a free group and its subgroup.
result Lower central subgroups of a subgroup do not contain those of the free group if the subgroup does not normally generate the free group.
The paper explores geometric finiteness in mapping class groups and constructs new examples of these subgroups.
problem Understanding geometric finiteness in mapping class groups and constructing new examples.
method Examined several constructions of subgroups and determined conditions for geometric finiteness.
result Provides new examples of parabolically geometrically finite and reducibly geometrically finite subgroups.
Study on braid group quotients by congruence subgroups.
problem Understanding the image of congruence subgroups in GL(n,Z).
method Characterization through symplectic congruence subgroups.
result Open problem solved: image of congruence subgroups in GL(n,Z).
Proposes a new method for finding non-redundant, standout subgroups in numeric datasets.
problem Mining large numbers of redundant subgroups in numeric datasets.
method Dispersion-aware problem formulation based on MDL principle for subgroup set discovery.
result Empirically demonstrates SSD++ returns outstanding subgroup lists.
Proves Congruence Subgroup Property for two types of groups.
problem Proving Congruence Subgroup Property for specific groups.
method Elementary proof of Johnson filtration and geometric subsurface inclusions.
result Proves Congruence Subgroup Property for nilpotent quotients and subsurface subgroups.
New method constructs non-quasiconvex subgroups in hyperbolic groups.
problem Creating non-quasiconvex subgroups in hyperbolic groups.
method Using Stallings-like techniques on right-angled Coxeter groups (RACGs).
result Explicit examples of non-quasiconvex subgroups constructed.
Paper studies Wiman-Edge pencil and Wiman curve, providing uniformizations and modular interpretations.
problem Understanding the geometry and uniformization of the Wiman-Edge pencil and Wiman curve.
method Explicit uniformizations of the Wiman-Edge pencil and Wiman curve as quotients of the hyperbolic plane and arithmetic quotients.
result Explicit uniformizations and modular interpretations of the Wiman-Edge pencil and Wiman curve.
New techniques reveal subgroup properties in Coxeter groups.
problem Characterizing and understanding subgroups of right-angled Coxeter groups.
method Using cube complexes and Stallings-like techniques to study subgroups.
result Reflection and one-ended subgroups are quasiconvex.
Characterizes knotted subgroups of Lie groups and provides examples.
problem Defining and understanding knotted subgroups of Lie groups.
method Geometric equivalence, one-parameter subgroups, infinitesimal elements, canonical forms, spectrum analysis.
result Completely classified knotted subgroups of SL(2,R) and SL(3,R).