Complex captures group properties, invariant under quasi-isometry.
problem Classical properties of subgroups in a group pair.
method Introduces coset intersection complex to study group properties.
result Quasi-isometry invariance of coset intersection complex.
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.
Closed Lorentz 4-manifolds have finite isometry groups with a bounded abelian subgroup.
problem Understanding the structure of isometry groups of closed Lorentz 4-manifolds.
method Proving that any finite subgroup of the isometry group of a closed Lorentz 4-manifold has a bounded abelian subgroup.
result Finite isometry groups of closed Lorentz 4-manifolds have a bounded abelian subgroup.
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further dev…
Study projective properties of non-flat Lorentzian surfaces.
problem Characterize projective transformations of Lorentzian surfaces.
method Analyzes properties of non-flat tori and topologically finite surfaces.
result Non-flat tori have at most index 2 in their projective group.
Study of symmetries in 3D Lie groups, determining index and moduli space properties.
problem Understanding symmetries in 3D Lie groups and their moduli space.
method Computed full isometry groups of left-invariant metrics on 3D Lie groups.
result Determined index of symmetry and properties of moduli space.
The paper examines conditions for Lie groups of isometries in specific metric measure spaces.
problem Conditions for the isometry and measure-preserving isometry groups to be Lie groups.
method Analyzes various metric measure spaces including RCD*, CD, CD*, and MCP spaces.
result Necessary and sufficient conditions for the groups to be Lie groups are identified.
Clarifies metric properties on group power sets.
problem Properties of metric on group power sets.
method Analysis of Hausdorff-like metric induced from word length norm.
result Results on quasi-isometries between subspaces of power sets.
Analyzes isometries in metric Lie groups, proving their regularity and composition properties.
problem Analyzing isometries in metric Lie groups, especially Carnot groups.
method Analyticity of isometries and composition properties proved via Hilbert 5th problem and Riemannian case.
result Every isometry between connected and nilpotent metric Lie groups is a composition of a left translation and an isomorphism.
The paper classifies groups that can be isometry groups of infinite-genus hyperbolic surfaces.
problem Which groups can be realized as isometry groups of infinite-genus hyperbolic surfaces?
method Classification of isometry groups for infinite-genus 2-manifolds with no planar ends.
result There is an uncountable class of 2-manifolds where every countable group can be realized as an isometry group.
The study characterizes geodesic orbit Riemannian spaces and their properties.
problem Characterizing geodesic orbit Riemannian spaces and their properties.
method Analyzing the structure of nilradical and radical of Lie algebra, discussing compact Lie group representations.
result Described the structure of nilradical and radical of Lie algebra of isometry group.
Study differential properties of symmetric real matrices with a specific metric.
problem Differential geometric properties of symmetric real matrices.
method Trace metric on non-singular symmetric real matrices, isometries of positive definite matrices.
result Description of the full group of isometries for positive definite matrices.
The Tits alternative applies to groups acting on specific CAT(0) spaces.
problem Understanding the structure of groups acting on CAT(0) spaces.
method Proving the Tits alternative for groups acting on visibility CAT(0) spaces with bounded packing property.
result Groups either almost nilpotent or contain a free nonabelian subgroup of rank 2.
Unified framework for generalized sparsity and RIP analysis.
problem Analyzing inverse problems with sparsity models.
method Proposed generalized notions of sparsity and a unified RIP framework.
result Extends RIP analysis to broader contexts including tensor products.
Study shows rigidity in cusp-decomposable manifolds' geometry.
problem Understanding the geometry of cusp-decomposable manifolds.
method Examined large scale geometry and quasi-isometries.
result Proved quasi-isometric rigidity for fundamental groups.
Improves data recovery with optimized measurements and generalized sparsity models.
problem Data recovery with optimized measurements and generalized sparsity models.
method Optimizing over families of Banach spaces, investigating preservation of difference of sparse vectors, extending RIP to group structured measurements, and extending Fourier measurement concepts to infinite dimensions.
result Optimal scaling of number of measurements for group structured measurements and improved RIP in infinite dimensions.
Geometric group theory explores groups through their geometric properties.
problem Understanding groups via geometric properties.
method Cayley and Schreier graphs, ping-pong lemma, quasi-isometries, growth of groups, hyperbolicity.
result Gromov's theorem on groups of polynomial growth and amenability.
As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like propertie…
Introduces halo products and studies their geometric properties.
problem Understanding the large-scale geometry of halo groups.
method Introduces halo products and builds a geometric framework.
result Provides refined invariants distinguishing halo groups up to quasi-isometry.
We show that if a group G acts by isometries on a metric space M which has asymptotic property C, such that the quasi-stabilizers of a point x∈M have asymptotic dimension less than or equal to n, then G itself has asymptotic property C.
Study metrics on half plane with specific curvature properties.
problem Warped product metrics on half plane.
method Holomorphic isometries and sectional curvature analysis.
result Metrics with zero and unbounded negative curvature exist.
Study on discrete properties of complex hyperbolic triangle groups.
problem Discreteness of complex hyperbolic triangle groups of type [m1, m2, 0].
method Analysis of isometries generated by complex reflections in ultra-parallel geodesics.
result Proves discreteness and non-discreteness results for these groups.
We say that a group has property R∞ if any group automorphism has an infinite number of twisted conjugacy classes. Fel'shtyn and Goncalves prove that the solvable Baumslag-Solitar groups BS(1,m) have property R∞. We define a solvable generalization Γ(S) of these groups which we show to have proper…
Uniformly perfect Morse boundaries characterize geometric properties of groups.
problem Characterizing geometric properties of groups using Morse boundaries.
method Introducing and geometrically characterizing uniformly perfect Morse boundaries for proper geodesic metric spaces.
result The Morse boundary of any finitely generated, non-elementary group is uniformly perfect if it is nonempty.
Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.
problem Understanding the algebraic, geometric, and topological properties of grid cells.
method Investigating recurrent neural network models of grid cells, focusing on Lie group and Lie algebra representations, conformal isometry, and hexagon periodic patterns.
result Conformal isometry leads to hexagon periodic patterns in grid cell responses and accurate path integration.
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determi…
Study horospheres in Teichmüller space and prove mapping class group properties.
problem Characterize horospheres in Teichmüller space and their relation to mapping class group.
method Analyze geometry of horospheres, prove properties of diffeomorphisms preserving horospheres, use metric ball relation.
result Every C1-diffeomorphism preserving horospheres is an element of the extended mapping class group. New classes of RAAGs have quasi-isometry coinciding with commensurability.
problem Quasi-isometry classification of RAAGs with infinite outer automorphism groups.
method Deformation argument and cubulation techniques.
result For certain RAAGs, quasi-isometry implies commensurability.
Algorithm for Teichmüller isometries induced by periodic mapping classes
problem Determining isometries in Teichmüller space induced by periodic mapping classes
method Algorithm using Fenchel-Nielsen coordinates
result Description of isometry induced by periodic mapping class of order 4g+2 The study shows rigidity properties are lost in hyperbolic generalizations.
problem Lack of geometric and topological rigidity in hyperbolic groups.
method Observations and proofs on acylindrically hyperbolic and relatively hyperbolic groups.
result No well-defined limit set for acylindrical actions on hyperbolic spaces.
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…
The study classifies Heintze groups up to isometry and quasi-isometry in low dimensions.
problem Classifying Heintze groups up to isometry and quasi-isometry in low dimensions.
method Analyzing quasi-isometries and isometries of Heintze groups, applying existing tools to groups of dimension 4 and 5.
result Complete classification of simply connected solvable groups in dimension 4 and groups of polynomial growth in dimension 5 up to isometry.
Study finds all isometries for specific Lie groups.
problem Identifying isometry groups in nonunimodular Lie groups.
method Examined left-invariant Riemannian metrics on Lie groups of dimension four.
result Determined full group of isometries for each metric.
Compact Lie groups have compact isometry groups with pseudo-Riemannian metrics.
problem Characterizing isometry groups of compact Lie groups with pseudo-Riemannian metrics.
method Analyzing left-invariant pseudo-Riemannian metrics on compact Lie groups.
result Isometry groups of compact Lie groups are compact.
Proves properties of arithmetic lattices and hyperbolic manifolds.
problem Properties of arithmetic lattices and hyperbolic manifolds.
method Study of normalizers of lattices and subgroup growth theory.
result Every arithmetic lattice has the property of being the normalizer of many sublattices.
Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and ε-spaces exhaust the class of n-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least 1/2n(n−1)+1, for almost all val…
Computes Weyl group of Kähler toric manifold isometries.
problem Computing the Weyl group of Kähler toric manifold isometries.
method Analyzes the group of holomorphic isometries of a Kähler toric manifold with real analytic Kähler metric.
result Computed the Weyl group of the group of holomorphic isometries.
Study proves rigidity of marked length spectra in contracting group actions.
problem Rigidity of marked length spectra in contracting group actions.
method Unified approach using the Extension Lemma and metric geometry.
result Orbit map is a rough isometry if marked length spectra match.
Study describes isometry groups of specific Lie groups.
problem Understanding isometry groups of compact Lie groups.
method Analyzing bi-invariant metrics on SO(4) and U(n).
result Full groups of isometries identified for specific Lie groups.
Study reveals structure of isometry group for specific manifolds.
problem Understanding the isometry group of non-compact, homogeneous manifolds.
method Analyzes non-compact, homogeneous manifolds with immortal Ricci flows.
result Establishes structure result for isometry group.
For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any density less than a half (or in the few relators model) contains quasiconvex subgro…
The study describes quasiflats in 2D Artin groups and their properties.
problem Understanding the structure and properties of quasiflats in 2D Artin groups.
method Metric systolicity and combinatorial analysis of tilings.
result Precise description of building blocks (atomic sectors) for quasiflats in 2D Artin groups.
Quantum isometry groups exist for certain metric spaces.
problem Existence of quantum isometry groups for specific metric spaces.
method Proved existence for geodesic metrics and uniformly distributed measure spaces.
result Quantum isometry groups are classical (commutative) for Riemannian manifolds.
Explicit isometry groups found for nearly Kähler manifolds.
problem Understanding the symmetries of nearly Kähler manifolds.
method Alternative, less algebraic approach to find isometry groups.
result Explicit expression for isometry groups of six-dimensional nearly Kähler manifolds.
New groups algebraically fibre with high-dimensional hyperbolic groups.
problem Finding new quasi-isometry classes of hyperbolic groups.
method Constructing infinitely many hyperbolic groups as finite-index subgroups of right-angled Coxeter groups.
result Groups algebraically fibre with finitely presented kernels, expanding finiteness properties.
Generalizes Bestvina's Z-boundaries to coarse Z-boundaries.
problem Establishing properties of Z-boundaries for groups. method Introducing a new concept of a 'coarse Z-boundary' and proving theorems about it. result Admitting a coarse Z-boundary is a pure quasi-isometry invariant. Authors define and study leaf space isometries of singular Riemannian foliations and their spectral properties.
problem The equality of specB(M1,F1) and specB(M2,F2) is not guaranteed by smooth isometry of leaf spaces. method The authors provide conditions under which the equality of specB(M1,F1) and specB(M2,F2) is guaranteed. result Additional geometric conditions on the leaves ensure the equality of specB(M1,F1) and specB(M2,F2). Sharp stability of isometries on Heisenberg group proven.
problem Quantitative stability of isometries on the Heisenberg group.
method Proving quasi-isometries close to isometries with specific closeness orders.
result Quasi-isometries of Heisenberg group are close to isometries with specific closeness orders.