Affine cactus groups are CAT(0) and hyperbolic.
problem Characterizing geometric properties of affine cactus groups.
method Analyzing CAT(0) and hyperbolic properties through group theory.
result Affine cactus groups of degree three are hyperbolic.
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48 results for “affine groups”
The paper explores flat affine and symplectic structures on Lie groups.
problem Exploring flat affine and symplectic structures on Lie groups.
method Left invariant affine structures, immersion of Lie groups, Koszul's method, Lagrangian bi-foliation.
result Flat left invariant affine symplectic connections and their associated affine symplectomorphisms.
Characterizes flat affine symplectic Lie groups and their properties.
problem Characterizing flat affine symplectic Lie groups.
method Using symplectic étale affine representations and central translations.
result Obtains nontrivial examples of flat affine symplectic Lie groups in every even dimension.
This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. Th…
Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
The paper characterizes flat affine connections on manifolds and Lie groups.
problem Characterizing flat affine connections on manifolds and Lie groups.
method New characterization through affine representations of automorphisms.
result Existence of a Lie group with a flat affine bi-invariant connection.
The affine-additive group is hyperbolic with a non-vanishing 4-capacity.
problem Characterizing the hyperbolicity of the affine-additive group.
method Proving local 4-Ahlfors regularity and hyperbolicity using a left-invariant metric and measure.
result The affine-additive group is hyperbolic with a non-vanishing 4-capacity.
New manifold structures on Weyl group orbit spaces proven.
problem Constructing generalized Frobenius manifold structures.
method Construction on orbit spaces of affine Weyl groups.
result Monodromy groups are parabolic subgroups.
Characterizes flat affine connections on manifolds.
problem Understanding flat affine connections on manifolds.
method Characterization through natural affine representation of diffeomorphisms.
result Group of affine transformations acts on R^n with open orbit when dimension > n.
Constructs proper affine actions for right-angled Coxeter groups.
problem Proper affine actions for right-angled Coxeter groups.
method Constructs proper actions of right-angled Coxeter groups on O(p,q+1) and its Lie algebra by affine transformations.
result Any virtually special group admits proper affine actions on some R^n.
This research studies affine invariance in continuous-domain convolutional neural networks.
problem Recognizing patterns and features under affine transformations in continuous domains.
method Introduces a new criterion for assessing affine invariance, embeds images into the affine Lie group, and analyzes convolution over this group.
result Extends the scope of geometrical transformations that deep-learning pipelines can handle.
New definitions and properties for hyperbolic group representations.
problem Understanding representations of hyperbolic groups into affine groups.
method Defining affine Anosov representations and proving their equivalence to Anosov linear parts with proper action.
result Affine Anosov representations of hyperbolic groups have specific properties related to their linear parts and proper actions.
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study th…
Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
Surface groups can't act properly on affine space if their linear part is Hitchin.
problem Proper affine actions of surface groups with specific linear parts.
method Independent proof of a theorem by Danciger and Zhang.
result Surface groups with Hitchin linear part cannot act properly on affine space.
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.
Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski planemath.DG
The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.
problem Computing curvature limits in affine and Minkowski groups.
method Analyzing Euclidean -smooth surfaces and curves in affine and Minkowski groups.
result Gauss-Bonnet theorems in affine and Minkowski groups are proven.
The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.
problem Reconstructing planar curves with specified Euclidean or affine curvatures.
method The paper presents algorithms for curve reconstruction under the special Euclidean and equi-affine groups.
result The reconstructed curves are close to the original curves in terms of the specified curvatures.
Proof of conjecture for affine Artin groups.
problem Asphericality of complements of affine hyperplane arrangements.
method Combinatorics of noncrossing partition posets, dual Artin groups, and topological models.
result Affine Artin groups are aspherical.
Affine Artin groups have a finite classifying space.
problem Proving the conjecture for affine Artin groups.
method Dual Garside structures, Euclidean isometries, and shellability of noncrossing partitions.
result Affine Artin groups have a finite classifying space.
The paper classifies compact affine quaternionic curves and surfaces.
problem Classifying compact affine quaternionic curves and surfaces.
method Affine quaternionic manifolds, Kodaira Theorem, fundamental groups, Lie Groups.
result Only quaternionic tori and primary Hopf surface S^3 x S^1 are compact affine quaternionic curves.
Constructs generalized Frobenius manifolds for specific Weyl groups.
problem Creating structures for orbit spaces of Weyl groups.
method Applying a previously established construction method to specific Weyl groups.
result Generalized Frobenius manifold structures constructed for and .
Paper generalizes connections between Lie groups and affine connections.
problem Exploring properties of infinitesimal groups and affine connections.
method Introducing second-order infinitesimal groups and using them to define Lie brackets and connections.
result Generalized correspondence between symmetric and non-symmetric affine connections.
New method constructs proper affine actions of groups in higher dimensions.
problem Finding proper affine actions of discrete groups in higher-dimensional spaces.
method Higher strip deformations and Margulis invariant for properness.
result Affine actions of convex cocompact groups and virtually free groups are constructed properly.
Real Lie groups' invariant theory matches that of their affine counterparts.
problem Matching invariant theory of real Lie groups with affine groups.
method Simple remark showing coincidence.
result Invariant theory of real Lie groups equals that of their affine counterparts.
The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.
problem Understanding cohomological dimensions of affine manifolds with specific holonomy groups.
method Analyzing the tangent bundle structure and using coarse geometry techniques.
result The cohomological dimension is bounded by the dimension minus the index of the holonomy group.
This paper classifies reversible and strongly reversible elements in affine groups.
problem Classifying reversible and strongly reversible elements in affine groups.
method Identifying affine transformations and using conjugacy by involutions.
result Classification of reversible and strongly reversible elements in affine groups.
We define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of . We prove a fuchsian affine action of a surface group is never proper.
The study of which mapping class group elements can be realized as affine automorphisms of dilation surfaces.
problem Which elements of the mapping class group can be realized as affine automorphisms of dilation surfaces?
method Investigation into the affine automorphism groups of dilation surfaces, including the construction of dilation surfaces from multicurves.
result Only certain types of mapping class group elements can arise as affine automorphisms of dilation surfaces.
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence questio…
The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
problem Computing curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
method Sub-Riemannian limits of Gaussian curvature, Schouten-Van Kampen affine connections, and adapted connections.
result Gauss-Bonnet theorems associated with Schouten-Van Kampen affine connections in the Heisenberg group.
Affine links in projective space have a specific group property.
problem Characterizing links in projective space.
method Examined the fundamental group of link complements.
result Affine links have a non-trivial element of order two in their fundamental group.
We prove a long-standing conjecture about complex reflection arrangements.
problem The conjecture for affine Artin groups.
method Recent advancements in dual Coxeter and Artin groups theory, new constructions, and poset shellability.
result The complexified complement of an affine reflection arrangement is a classifying space.
The paper studies Anosov holonomy groups in complete affine manifolds.
problem Characterizing Anosov holonomy groups in complete affine manifolds.
method Representation theory and coarse geometry techniques.
result Complete affine manifolds with Anosov holonomy groups have specific geometric properties.
The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
problem Computing curvature and geodesic curvature for surfaces and curves in affine and rigid motions groups.
method Defined deformed Schouten-Van Kampen connections, computed Gaussian curvature limits, and signed geodesic curvature.
result Derived Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
Affine 3-manifolds with centralizing holonomy are complete.
problem Closed flat affine manifolds with parallel volume are not always complete.
method Showed the Markus conjecture holds for 3-manifolds with centralizing holonomy.
result Holonomy centralizing an affine transformation preserves volume completeness.
Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …
Notes on flat pseudo-Riemannian manifolds, focusing on their characterization and properties.
problem Characterizing flat pseudo-Riemannian manifolds and their properties.
method Survey of basic concepts in affine and Riemannian geometry, characterization of flat manifolds, and analysis of Lie groups.
result Characterization and properties of flat pseudo-Riemannian Lie groups and their metrics.
The paper classifies vector fields on 5D nilpotent Lie groups.
problem Classifying left-invariant affine and projective vector fields on 5D nilpotent Lie groups.
method Algebraic characterization and case-by-case analysis of vector fields.
result All projective vector fields are affine, extending classical results.
The complete classification of representations of the Trefoil knot group G in S^{3} and SL(2,R), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the non cyclic groups of affine Euclidean isometries generat…
Affine diffeomorphisms are undistorted in half-translation surfaces.
problem Undistorted nature of affine diffeomorphism groups.
method Proof of undistorted subgroup property and systole map embedding.
result Finitely generated subgroups of affine diffeomorphisms are undistorted.
In his 1990 doctoral thesis, Todd Drumm showed that proper affine deformations of free Fuchsian groups could be constructed as Schottky groups using a new family of hypersurfaces called "crooked planes." The existence of proper affine deformations of Fuchsian Schottky groups was demonstrated by Margulis in the early 19…
Generalizes Lee's result to virtually polycyclic groups.
problem Understanding morphisms between virtually polycyclic groups.
method Study of translation-like actions and their behavior under subgroups and cosets.
result Generalization of Lee's result to virtually polycyclic groups.
Let be an ordered abelian group. We show how an group -- that is, a group admitting a free affine action without inversions on a -tree -- admits a natural graph of groups decomposition, where vertex groups inherit actions on -trees. Using recent work o…
The goal of this paper is to provide a method, based on the theory of extensions of left-symmetric algebras, for classifying left-invariant affine structures on a given solvable Lie group of low dimension. To better illustrate our method, we shall apply it to classify complete left-invariant affine structures on the os…
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the con…
Researchers study rational and pretzel knots using affine group representations.
problem Understanding the structure and properties of rational and pretzel knots.
method Constructing representations of knot groups into the affine group AGL(1,ℂ) via a TQFT valued in spans of singular vector bundles.
result Closed-form expressions for Alexander polynomials and bounds on their zeros for rational and pretzel knots.