The paper classifies obstructions to real Clifford module bundles and their relation to spin structures.
problem Topological obstructions to real Clifford module bundles over pseudo-Riemannian manifolds.
method Analyzes real Lipschitz structures and their relation to spin structures.
result Classifies real Lipschitz structures in all dimensions and signatures.
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. We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Ru…
Study maps in semidirect products of groups, proving Lipschitz properties without intrinsic dilations.
problem Proving Lipschitz conditions in semidirect products of groups without intrinsic dilations.
method Using equivalent conditions and properties of projection maps in metric spaces.
result Proves the same Lipschitz results as in Carnot groups, without intrinsic dilations.
Corrects omissions in a paper about Lipschitz connectivity and invariants.
problem Lipschitz connectivity and filling invariants in solvable groups and buildings
method None specified in the abstract, focuses on correcting omissions
result Corrected omissions in a previous paper
Novikov conjecture reduced to Lipschitz cohomology of groups.
problem Novikov higher signature conjecture for groups.
method Introducing Lipschitz cohomology classes and reducing the conjecture.
result Reduced Novikov conjecture to Lipschitz cohomology.
Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.
problem Characterize the Lipschitz homotopy groups of contact 3-manifolds.
method Sub-Riemannian geometry, geometric measure theory, biLipschitz equivalence, purely unrectifiable sets.
result Contact 3-manifolds are K(π,1) spaces with uncountably generated first homotopy groups. Characterizes functions in Carnot groups of step 2.
problem Understanding intrinsic Lipschitz functions in Carnot groups.
method Characterization via intrinsic distributional gradients.
result Characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2.
The paper proves Rademacher's theorem for Heisenberg groups.
problem Proving Rademacher's theorem for Heisenberg groups.
method New definition of intrinsic Lipschitz graphs, extension and approximation theorems, use of Heisenberg currents and Rumin's complex.
result Rademacher's theorem for intrinsic Lipschitz graphs in Heisenberg groups.
Maps in Carnot groups are equivalent to solutions of a PDE system.
problem Understanding maps in Carnot groups of step 2.
method Equivalence between intrinsic Lipschitz maps and solutions to a PDE system.
result Intrinsic Lipschitz maps are equivalent to weak solutions of a PDE system.
The study proves surfaces in a specific Heisenberg group must be simple planes.
problem Characterizing surfaces in a sub-Finsler Heisenberg group.
method Analyzes (X,Y)-Lipschitz surfaces in H1 with a sub-Finsler structure. result Complete, oriented, stable (X,Y)-Lipschitz surfaces are vertical planes. Odd-dimensional Heisenberg groups can't extend maps Lipschitzly.
problem Lipschitz extension in Heisenberg groups.
method Analyzing metric spaces $(\H^n, \H^n)$ for odd n. result No Lipschitz extension property for odd n. Lipschitz and horizontal maps from an n-dimensional space into the (2n+1)-dimensional Heisenberg group $\H^n$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from Sk to $\H^n$ which factor through n-spheres and sh…
Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.
problem Regularity of boundaries in plentiful groups.
method Lipschitz approximation of boundaries.
result Boundary of almost minimizers can be approximated by intrinsic Lipschitz graphs.
Let ρ:G→O(V) be a real finite dimensional orthogonal representation of a compact Lie group, let σ=(σ1,…,σn):V→Rn, where σ1,…,σn form a minimal system of homogeneous generators of the G-invariant polynomials on V, and set $d = \max_i \operatorname{deg} …
Proves measure contraction for specific sub-Riemannian structures.
problem Measure contraction properties in sub-Riemannian structures.
method Analytic sub-Riemannian structures and Lipschitz Carnot groups.
result Proves measure contraction properties for the structures.
The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.
problem Extending isometry properties of Lipschitz metric to virtually free groups.
method Analyzing finite-index subgroups and their covers, identifying folding paths, and using deformation retraction.
result Existence of candidates for Lipschitz distance and deformation retraction of spine.
Existence and rigidity results for lifts in Carnot groups.
problem Existence and properties of lifts for maps between Carnot groups.
method Use central extensions to define lifts and prove existence and rigidity results for Lipschitz, Sobolev, and quasiconformal maps.
result Quasiconformal maps admit contact lifts that are bi-Lipschitz.
Investigates intrinsic Lipschitz sections in nonlinear quotient maps.
problem Analyzing intrinsic Lipschitz sections in non-linear quotient maps.
method Introduced Leibniz formula for intrinsic slope under weaker conditions, used properties of intrinsic dilations in Carnot groups, and provided conditions for sum of sections.
result Found conditions for sum of intrinsically Lipschitz sections in Carnot groups of step 2.
The study proves intrinsic graphs in Heisenberg group are planes if they meet certain conditions.
problem Characterizing intrinsic graphs in Heisenberg group with specific properties.
method Analyzing graphs with Lipschitz continuity and sub-Riemannian area variations.
result Intrinsic graphs in Heisenberg group are planes if they have zero first variation and non-negative second variation.
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
Study invariant Lipschitz bandits, improving regret bounds.
problem Optimizing decisions under symmetry in online settings.
method Integrates side observations using group orbits into UniformMesh algorithm.
result Improved regret bound for invariant Lipschitz bandit class.
We prove that Lipschitz intrinsic graphs in the Heisenberg groups Hn, with n>1, which are vanishing viscosity solutions of the minimal surface equation are smooth.
Study shows arc and curve graphs are retracts of free group splitting complexes.
problem Understanding free group splittings through arc and curve graphs.
method Proved arc and curve graphs are coarse Lipschitz retracts of free splitting complexes and other related graphs.
result Arc and curve graphs are retracts of free group splitting complexes.
Authors create a biLipschitz embedding of spheres into jet space Carnot groups without Lipschitz extensions.
problem Embedding spheres into jet space Carnot groups without Lipschitz extensions.
method Constructing a biLipschitz embedding of Sn into Jk(Rn) and proving lack of Lipschitz extension. result Embedding of spheres into jet space Carnot groups does not admit Lipschitz extensions.
Characterizes hypergenerated stratified groups with flat boundaries.
problem Characterizing stratified groups with flat boundaries.
method Algebraic characterization and embedding analysis.
result Hypergenerated groups have locally bi-Lipschitz embeddings of non-characteristic hypersurfaces.
We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with …
Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.
problem Proving Lipschitz continuity of sub-elliptic harmonic maps between singular spaces.
method Analyzing sub-elliptic harmonic maps from the Heisenberg group into CAT(0) spaces.
result Local Lipschitz continuity established for sub-elliptic harmonic maps.
We give some new methods, based on Lipschitz extension theorems, for bounding filling invariants of subsets of nonpositively curved spaces. We apply our methods to find sharp bounds on higher-order Dehn functions of Sol_{2n+1}, horospheres in euclidean buildings, Hilbert modular groups, and certain S-arithmetic groups.
Introduces intrinsically Lipschitz graphs in metric spaces.
problem Graphs in metric spaces with Lipschitz conditions.
method Focuses on quotient maps and intrinsically Lipschitz sections.
result Compactness, Ahlfors regularity, and extension theorems.
We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, πmLip(Hn), in terms of properties of the classical homotopy group of the sphere, πm(Sn). As an application we provide a new simplified proof of the fact that πnLip(Hn)=0, n=1,2,..., a…
We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg group with its Carnot-Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem.
We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S^n,Y) are not dense in the Sobolev space W^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz …
We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in sense of Connes-Gromov-Moscovici [CGM]. The conditions are stated in terms of the Stone-Čech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the suf…
Extends Lipschitz functions while preserving local constants.
problem Extending Lipschitz functions on metric spaces while maintaining local constants.
method Extends Lipschitz functions on metric spaces while locally preserving the asymptotic Lipschitz constant.
result Sobolev spaces on metric measure spaces are invariant under isomorphism of mm-structures.
The abstract discusses the classification of 3D Lie groups with Riemannian metrics.
problem Classifying 3D Lie groups up to quasi-isometries and bi-Lipschitz equivalence.
method Review of existing literature and study of quasi-isometry and bi-Lipschitz equivalence.
result For three-dimensional simply connected groups, quasi-isometry implies isometry with suitable metrics.
The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median su…
Article shows AdS-quasi-Fuchsian groups' limit sets are not smooth.
problem Smoothness of limit sets of AdS quasi-Fuchsian groups.
method Analyzes limit sets of AdS quasi-Fuchsian groups in PO(n,2).
result Limit sets are never C^1, except for Fuchsian groups.
Nilpotent groups can't be biLipschitz embedded into L1.
problem Proving that simply connected nilpotent Lie groups cannot be biLipschitz embedded into L1. method Using a pull-back distance and cut measures, the authors show that bi-Lipschitz embeddings can't exist in non-abelian settings.
result Every Carnot group that biLipschitz embeds into L1 is abelian. The paper studies maps in the Heisenberg group and their images, called Rickman rugs.
problem Understanding maps and their images in the Heisenberg group.
method Analyzes maps f:WoH, where H is the first Heisenberg group and W is a vertical subgroup. result Rickman rugs in the Heisenberg group admit a corona decomposition by intrinsic bilipschitz graphs.
For a finitely generated group G, we introduce an asymmetric pseudometric on projectivized deformation spaces of G-trees, using stretching factors of G-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in t…
Extends spinor-horosphere correspondence to higher dimensions and new spinor types.
problem Constructing isomorphisms between spinors and horospheres in hyperbolic spaces.
method Generalizes Mathews' results to Lipschitz spinors and null multiflags in higher-dimensional hyperbolic spaces.
result Equivariant correspondence between two-component Lipschitz spinors and higher-dimensional horospheres.
Reduced sample complexity for group-invariant distributions.
problem Improving sample complexity for estimating divergences of group-invariant distributions.
method Quantified reduction in sample complexity for Wasserstein-1 metric and Lipschitz-regularized α-divergences under finite and infinite groups.
result Sample complexity reduction proportional to group size for finite groups, and convergence rate depends on intrinsic dimension for infinite groups.
In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φ_y+ [φ^{2}/2]_t=w, where w is a bounded function depending o…
This paper proves metric rectifiability for certain Heisenberg group surfaces.
problem Equivalence of rectifiability definitions for Heisenberg group surfaces.
method New criterion for finding bilipschitz maps between metric spaces.
result Metric bilipschitz rectifiability for α-Hölder continuous surfaces in Hn. The paper explores rectifiability in sub-Riemannian geometry, finding a smooth hypersurface with unique properties.
problem Finding a good notion of rectifiability in sub-Riemannian geometry, focusing on smooth hypersurfaces.
method Study of a specific smooth hypersurface in Carnot groups, analyzing its tangent groups and rectifiability properties.
result The existence of a C∞ hypersurface with uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number.
Let Gamma_0 be a discrete group. For a pair (j,rho) of representations of Gamma_0 into PO(n,1)=Isom(H^n) with j geometrically finite, we study the set of (j,rho)-equivariant Lipschitz maps from the real hyperbolic space H^n to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic l…