Computes cohomology groups for NEC groups, focusing on Fuchsian groups.
problem Understanding the cohomology of non-Euclidean crystallographic groups.
method Computes cohomology groups for geometrically finite NEC groups, and determines the ring structure for Fuchsian groups.
result Determination of cohomology groups and ring structures for Fuchsian groups.
Abstract: Surveying K-theory for Farrell-Jones Conjecture on Euclidean retracts.
problem Farrell-Jones Conjecture for K-theory.
method Actions on Euclidean retracts.
result Applications to GL_n(Z), relative hyperbolic groups, and mapping class groups.
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.
Study groups of piecewise isometries in tessellations of Euclidean space.
problem Understanding the structure of groups formed by cutting and gluing tessellations.
method Proving structure results about groups of piecewise isometries of tessellations, including elementary amenability.
result Groups of piecewise isometries of tessellations are elementary amenable.
Paper proves inequalities for differential forms in Euclidean and Heisenberg groups.
problem Proving inequalities for differential forms in limiting L^1 norm.
method Interior Poincaré and Sobolev inequalities, replacing L^p estimates.
result Developed inequalities for L^1 norm, including singular integral estimates.
The geometry of conjugation is mapped within Euclidean isometry groups.
problem Understanding conjugacy classes and their transformations in Euclidean groups.
method Geometric description of conjugacy classes and sets of conjugating elements based on linearizations.
result The conjugacy classes and sets of conjugating elements are described by the move-set and fix-set of linearizations.
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
New growth rate identified for quaternionic Heisenberg group filling functions.
problem Identifying the growth rate of quaternionic Heisenberg group filling functions.
method Analyzing the growth of filling volume functions in the quaternionic Heisenberg group up to dimension n+1.
result Strictly faster growth rate identified for dimension n+1 compared to Euclidean space.
Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.
problem Understanding fundamental groups of open manifolds with specific curvature properties.
method Analyzing universal covers with Euclidean volume growth and applying topological group theory.
result Fundamental groups are finitely generated and virtually abelian under given conditions.
Groups act on Euclidean buildings with a minimum dimension bound.
problem Understanding the minimum dimension a group can act on properly.
method Analyzing actions on Euclidean buildings and S-arithmetic groups.
result Action dimension is at least twice the building's dimension.
In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two generator real Mobius group acting on the Poincare plane is or is not discrete can be interpreted as a non-Euclidean Euclidean algorithm. That is, the algorithm can be viewed as an application of the Euclidean division alg…
Geometric bijection and homotopy equivalence between Lie group orbits.
problem Isomorphic adjoint and coadjoint representations of Lie groups.
method Geometric bijection and homotopy equivalence of orbits.
result Geometrically defined bijection and homotopy equivalence between adjoint and coadjoint orbits.
The paper presents presentations for Euclidean Picard modular groups.
problem Finding presentations for arithmetic non-cocompact lattices in Euclidean domains.
method Applying Macbeath's theorem to a Γ-invariant covering by horoballs.
result Presentations for Picard modular groups with d=2,11 are obtained.
Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.
problem Computing Laplace-Beltrami on constrained submanifolds.
method Embedded gradient vector field method, explicit formula derivation.
result Explicit formula for Laplace-Beltrami on orthogonal group.
Formula derived for volume entropy of certain metrics on Euclidean space.
problem Volume entropy calculation for a family of metrics.
method Derived a formula for volume entropy of metrics in a family of generalized SOL and hyperbolic space metrics.
result Solved a conjecture related to a family of 3-manifolds.
The goal of this article is to show that five explicitly given transformations, a rotation, two screw Heisenberg rotations, a vertical translation and an involution generate the Euclidean Picard modular groups with coefficient in the Euclidean ring of integers of a quadratic imaginary number field. We also obtain the r…
We show that a well known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group.
Proves regularity of harmonic maps into Euclidean buildings and applies to superrigidity of algebraic groups.
problem Regularity of harmonic maps into Euclidean buildings.
method Analyzes singular sets and applies geometric settings.
result Proves singular sets of Hausdorff codimension 2 for harmonic maps.
Weakly-irreducible not irreducible subalgebras of $\so(1,n+1)$ were classified by L. Berard Bergery and A. Ikemakhen. In the present paper a geometrical proof of this result is given. Transitively acting isometry groups of Lobachevskian spaces and transitively acting similarity transformation groups of Euclidean spaces…
The paper proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
problem Properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
method Proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth are finitely generated.
Study shows no new Euclidean factors can appear in the limit of CAT(0) spaces.
problem Stability of Euclidean factors in CAT(0) spaces under convergence.
method GH-convergence of CAT(0) spaces with uniformly cocompact discrete groups of isometries.
result Dimension of the maximal Euclidean factor is the same for large j. This study classifies noncompact quasi-Einstein manifolds conformal to Euclidean spaces.
problem Investigating nontrivial quasi-Einstein manifolds globally conformal to Euclidean spaces.
method Considering manifolds with invariant conformal factors and potential functions under an (n−1)-dimensional translation group. result Complete classification of manifolds when λ=0 and m≥1 or m=2−n. Non-Euclidean number rings have non-integral Steinberg modules.
problem Characterizing when Steinberg modules are generated by integral elements.
method Analyzing special linear groups over non-Euclidean imaginary number rings.
result Steinberg modules are not generated by integral elements in non-Euclidean rings.
Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.
problem Minkowski type problem in Heisenberg groups.
method Variational method.
result Positive answer to sub-Riemannian Minkowski type problem.
The paper provides bounds for embedding manifolds into Euclidean spaces with group actions.
problem Finding explicit bounds for embedding manifolds into Euclidean spaces with group actions.
method The paper provides upper and lower bounds for the dimension of the Euclidean space required for equivariant embeddings of manifolds into Euclidean spaces for finite group actions.
result Explicit bounds for the dimension of the Euclidean space required for equivariant embeddings of manifolds into Euclidean spaces for finite group actions.
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.
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete classification. These groups naturally arise in the study of the quotient of a Euclidean spa…
Computes group of ring motions for a specific link structure.
problem Computing the group of motions for a specific type of link.
method Study of a short exact sequence of groups of ring motions for general ring links in R^3.
result Builds a presentation for the group of motions of H-trivial links with an arbitrary number of components.
This paper classifies Lie groups embeddable into 4D space.
problem Determining the minimum dimensional Euclidean space for Lie group embeddings.
method Used algebraic equations (Gauss and derived Gauss) to classify Lie groups.
result Classified Lie groups embeddable into 4D space.
Investigates fundamental groups and path lifting for algebraic varieties.
problem Understand fundamental groups and path lifting properties of algebraic varieties.
method Analyzes three fundamental questions about fundamental groups of algebraic varieties.
result Identifies conditions for surjectivity on fundamental groups and path lifting properties.
This paper covers non-orientable Euclidean manifolds B3 and B4, detailing their n-fold coverings.
problem Describing and calculating n-fold coverings of non-orientable Euclidean manifolds B3 and B4.
method Classifying subgroups in the fundamental groups π1(B3) and π1(B4) up to isomorphism, calculating numbers of subgroups and conjugacy classes for each isomorphism type.
result The numbers of non-equivalent n-fold coverings of each type of B3 and B4 are determined.
The paper proves bounded cohomology properties of Euclidean space groups.
problem Understanding bounded cohomology of transformation groups of Euclidean spaces and discs.
method Analyzes groups of homeomorphisms and diffeomorphisms of Euclidean spaces and discs, proving boundedly acyclic properties.
result Groups of orientation-preserving homeomorphisms and diffeomorphisms of Rn are boundedly acyclic, with implications for characteristic classes of bundles. When Daan Krammer and Stephen Bigelow independently proved that braid groups are linear, they used the Lawrence-Krammer-Bigelow representation for generic values of its variables q and t. The t variable is closely connected to the traditional Garside structure of the braid group and plays a major role in Krammer's alge…
Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.
problem Identifying metric lines in the Special Euclidean group on the plane.
method Alternative proof using Hamilton-Jacobi theory.
result Metric lines in SE(2) are identified.
Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of…
We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod p acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov …
Study introduces combinatorial criterion for quasi-isometry groups of Euclidean spaces.
problem Determining quasi-isometries of Euclidean spaces.
method Introduces PLδ-homeomorphisms and combinatorial criterion using vertices and edges of simplicial structures. result The center of the quasi-isometry group QI(Rn) is trivial. Study classifies special surfaces in space with translational and rotational symmetries.
problem Classifying surfaces with specific symmetries and densities.
method Analyzes λ-translating solitons with invariant properties under translations and rotations. result Classifies all λ-translating solitons with invariant surfaces. The study provides bounds for geodesic diameter in Euclidean space.
problem Finding bounds for geodesic diameter in Euclidean space.
method Develops a geometric approach using locally rectifiable chains and complete normed commutative group bundles.
result Provides a new method for calculating geodesic diameter bounds.
The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.
problem The non-increasing property of numbers of almost Euclidean factors of geodesic balls.
method Proves a transformation theorem under a non-decreasing property compared to the non-increasing property.
result Shows that for a manifold with nonnegative Ricci curvature, if its universal cover is polar at infinity and the number of almost Euclidean factors is monotone, then its fundamental group is finitely generated and virtually abelian.
Ricci flow emerges from renormalizing nonlinear Sigma models.
problem Understanding the renormalization group flow in nonlinear Sigma models.
method Using Euclidean algebraic quantum field theory and Wick ordering.
result The first-order renormalization group flow of nonlinear Sigma models equals the Ricci flow.
We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for …
Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption i…
Affine Artin groups have a finite classifying space.
problem Proving the K(π,1) 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.
Paper proves equivalence of derivatives for maps between Carnot groups.
problem Maps between Carnot groups and their derivatives.
method Elementary proof using Euclidean arguments and mean value estimates.
result Maps preserving horizontal curves are continuously Pansu differentiable.
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 C2-smooth surfaces and curves in affine and Minkowski groups. result Gauss-Bonnet theorems in affine and Minkowski groups are proven.
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
Study expanding gradient Ricci solitons with Euclidean base.
problem Characterize expanding gradient Ricci solitons with specific properties.
method Analyze warped products with Euclidean base and invariant warping functions.
result Derive complete examples of expanding gradient Ricci solitons.