arXiv research

The study classifies Heintze groups up to isometry and quasi-isometry in low dimensions.

Study finds all isometries for specific Lie groups.

Computes Weyl group of Kähler toric manifold isometries.

Study describes isometry groups of specific Lie groups.

Study reveals structure of isometry group for specific manifolds.

Explicit isometry groups found for nearly Kähler manifolds.

Sharp stability of isometries on Heisenberg group proven.

Lifts isometries in orbit spaces for compact groups.

Study of symmetries in 4D Lie groups.

The paper classifies isometries on specific Lie groups.

Measure-scaling quasi-isometries on graphs have specific scaling groups.

The study shows finite measure-preserving isometry groups for certain metric measure spaces.

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.

Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is compact if $G$ is not simply-connected.

Study of isometries in spacetimes without observer horizons.

Classifies certain graph 2-braid groups up to quasi-isometry.

Study quasi-isometry invariants of square complexes and their applications.

We classify isometries of 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.

Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.

The paper classifies groups that can be isometry groups of infinite-genus hyperbolic surfaces.

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.

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.

Previously one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincaré duality groups for each $n\geq 4$. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\geq 4$ there are uncountably many…

New concept SB-generation helps classify transformation groups.

Study introduces combinatorial criterion for quasi-isometry groups of Euclidean spaces.

Groups can act on $L^p$ spaces with non-trivial cohomology.

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.

Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.

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.

Survey on quasi-isometries of group pairs and their invariants.

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 $\mathrm{Diff}(\mathbb{S}^1)$ obtained are semi conjugate to subgroups of finite covers of $\mathrm{PSL}(2,\mathbb{R})$ by…

Study classifies special metrics for which the isometry group of 3D Lie groups is larger than expected.