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.
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 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.
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.
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.
Lifts isometries in orbit spaces for compact groups.
problem Isometries in orbit spaces of compact groups.
method Equivariant isometry of original Euclidean space.
result Simple formula for connected component of isometry group.
Study of symmetries in 4D Lie groups.
problem Understanding symmetries in specific Lie groups.
method Analyzing isometry groups of left-invariant metrics.
result Full description of isometry groups for 4D Lie groups.
The paper classifies isometries on specific Lie groups.
problem Classifying isometries on nonnilpotent, solvable 3D Lie groups.
method Proving automorphisms are the only isometries for rank two Almost-Riemannian Structures.
result A classification result for rank two ARSs on nonnilpotent, solvable 3D Lie groups.
Measure-scaling quasi-isometries on graphs have specific scaling groups.
problem Understanding the scaling groups of graphs under quasi-isometries.
method Analyzing measure-scaling quasi-isometries on graphs and their properties.
result The scaling group of a graph is invariant under measure-scaling quasi-isometries.
The study shows finite measure-preserving isometry groups for certain metric measure spaces.
problem Understanding the structure of isometry groups in metric measure spaces.
method Analyzing synthetic negative Ricci curvature and Bakry-Émery Ricci curvature.
result The measure-preserving isometry group is finite for compact metric measure spaces with specific curvature conditions.
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For sy…
For an Alexandrov space (with curvature bounded below), we determine the maximal dimension of its isometry group and show that the space is isometric to a Riemannian manifold, provided the dimension of its isometry group is maximal. We also determine a gap in the possible dimensions of the isometry groups and show that…
The study explores maximal symmetry in Ricci solitons on Lie groups.
problem Maximal symmetry in left-invariant Riemannian metrics and Ricci solitons.
method Analysis of left-invariant metrics and Ricci solitons on Lie groups, using tools from previous work on Einstein metrics.
result Expanding homogeneous Ricci solitons have maximal isometry algebras but not always maximal isometry groups.
Study of isometries in spacetimes without observer horizons.
problem Understanding the symmetries of spacetimes without specific boundaries.
method Analysis of isometry groups in causal spacetimes without observer horizons.
result The group of time orientation-preserving isometries acts properly on the spacetime.
Classifies certain graph 2-braid groups up to quasi-isometry.
problem Classifying 2-braid groups over graphs up to quasi-isometry.
method Using intersection complexes and right-angled Artin groups.
result Classifies 2-braid groups over graphs with circumference ≤ 1 up to quasi-isometry.
Study quasi-isometry invariants of square complexes and their applications.
problem Classifying quasi-isometry types of 2D right-angled Artin groups and graph 2-braid groups.
method Define and analyze intersection complexes for universal covers of weakly special square complexes.
result Discover new quasi-isometric relationships between graph 2-braid groups and right-angled Artin groups.
We classify isometries of compact Lorentz manifolds.
problem Understanding the isometry group of compact Lorentz manifolds.
method Proving structure theorems and applying the Tits alternative.
result Classification of lattices acting on compact Lorentz manifolds.
We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the secon…
Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our constructi…
Using an extension to isometries of the associated Sasaki structure, we establish a Lie transformation group structure for the set of isometries of a pseudo-Finsler conical metric.
This is Part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in Part I, a compactification of these isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a ``trivial '' compactification. Here we show a geometric rigidity of non-bi-polarized Lorentz manifolds; …
Classifies homogeneous hypersurfaces in specific 4D geometries.
problem Classifying homogeneous hypersurfaces in 4D Thurston geometries.
method Analyzing subalgebras of Lie algebras and isometry groups.
result Determined all homogeneous hypersurfaces up to ambient isometries.
Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.
problem Characterizing homogeneous Lorentzian three-manifolds with a 4D isometry group.
method Explicit global coordinate description and proof of Ricci soliton properties.
result All special examples are non-gradient expanding Ricci solitons.
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.
This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannain situation; for instance the action of the nilradical of the isometry group does not need to…
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.
This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.
We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.
Study on volume growth of horospheres in specific Heintze groups.
problem Volume growth of horospheres in diagonalizable Heintze groups.
method Analysis of volume growth in Heintze groups with diagonalizable structure.
result Two distinct isometry and quasi-isometry classes of horospheres with explicit volume growth calculations.
Previously one of the authors constructed uncountable families of groups of type FP and of n-dimensional Poincaré duality groups for each n≥4. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each n≥4 there are uncountably many…
New concept SB-generation helps classify transformation groups.
problem Classifying transformation groups through quasi-isometry invariants.
method Identifying SB-generated groups in specific transformation groups.
result SB-generation provides robust extension of finite generation.
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. Groups can act on Lp spaces with non-trivial cohomology.
problem Understanding actions of groups on Hadamard spaces.
method Analyzing isometric actions on Hadamard manifolds and Lp-spaces. result Non-trivial Lp-cohomology for certain groups. Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily non-unimodular. In this work we consider unimodular solvable Lie groups and pro…
A special group of transformations of the real line cannot act effectively on it.
problem Understanding the limitations of transformations on the real line.
method Analyzing the group of orientation-preserving quasi-isometries of the real line.
result The group of quasi-isometries of the real line cannot act effectively on the line.
Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.
problem Understanding the structure of diffeomorphism groups of 3-manifolds.
method Deformation retraction and earlier work by multiple authors combined.
result Homotopy type of Diff(X) determined for prime 3-manifolds.
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…
We show that the group of smooth isometries of a complemented sub-Riemannian manifold form a Lie group and establish dimension estimates based on the torsion of the canonical connection. We explore the interaction of curvature and the structure of isometries and Killing fields and derive a Bochner formula for Killing f…
Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X) when X is a simply connected, nonpositively curved metric spaces by considering…
We prove that the identity component of the holomorphic isometry group of a Sasaki-Einstein metric is the identity component of a maximal compact subgroup of its automorphism group.
In this paper, based on research on rank-one isometries by W.Ballmann and M.Brin and recent research on rank-one isometries of Coxeter groups by P.Caprace and K.Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter g…
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.
Survey on quasi-isometries of group pairs and their invariants.
problem Understanding quasi-isometries of group pairs and their invariants.
method Exploration of quasi-isometry and qi-characteristic collections of subgroups.
result New insights into phenomena observed in quasi-isometric rigidity.
We study the isometry group of a globally hyperbolic spatially compact Lorentz surface. Such a group acts on the circle, and we show that when the isometry group acts non properly, the subgroups of Diff(S1) obtained are semi conjugate to subgroups of finite covers of PSL(2,R) by…
Study classifies special metrics for which the isometry group of 3D Lie groups is larger than expected.
problem Classifying metrics for which the isometry group of 3D Lie groups is larger than expected.
method Lie-theoretical methods to classify pairs (G, g)
result Determines the dimension of the isometry group for every pair (G, g)