Let denote the complex hyperbolic space of dimension . The group acts as the group of isometries of . In this paper we investigate when two isometries of the complex hyperbolic space commute. Along the way we determine the centralizers.
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Maps on the Morse boundarymath.GT
The paper explores additional structures on Morse boundaries to distinguish hyperbolic spaces up to quasi-isometry.
problem Distinguishing hyperbolic spaces up to quasi-isometry using additional structures on Morse boundaries.
method Investigates additional structures on Morse boundaries and proves conditions for a homeomorphism to be induced by a quasi-isometry.
result A homeomorphism between Morse boundaries of hyperbolic spaces is induced by a quasi-isometry if and only if it is bihölder, quasi-symmetric, or strongly quasi-conformal.
It is of interest to characterize algebraically the dynamical types of isometries of the complex and quaternionic hyperbolic planes. In the complex case, such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of the isometries of the two-dimensional quate…
This study extends a result on quasi-isometry of hyperbolic groups to relatively hyperbolic groups.
problem Classifying groups up to quasi-isometry, focusing on relatively hyperbolic groups.
method Defining quasiconformal maps and showing their equivalence to coarsely cusp-preserving quasi-isometries between Bowditch boundaries.
result Quasiconformal maps between Bowditch boundaries of relatively hyperbolic groups are equivalent to coarsely cusp-preserving quasi-isometries.
Bi-geodesic mappings preserve distances on hyperbolic surfaces with boundaries.
problem Preserving distances on hyperbolic surfaces with boundaries.
method Proving bijections between geodesics are isometries.
result A bijection between geodesics is an isometry.
The study proves conditions for hyperbolic isometries on fine curve graphs of higher genus surfaces.
problem Conditions for hyperbolic isometries on fine curve graphs of higher genus surfaces.
method Proves equivalence of conditions involving isotopic maps, pseudo-Anosov maps, and ergodic rotation sets.
result Ergodic homological rotation sets have nonempty interior for certain isotopic maps.
We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere.
Study of isometries on hyperbolic 3-manifold cusps.
problem Understanding transitivity in hyperbolic 3-manifold actions.
method Analyzing multiply transitive actions of isometries on cusps.
result Proved a conjecture about the maximum transitivity and upper bounds on cusps.
Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.
problem Understanding the length of elements in PU(2,1) relative to special elliptic isometries.
method Generalizing the involution length of the complex hyperbolic plane, calculating the -length of PU(2,1) and describing decompositions of isometries.
result Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.
We prove that a PQ-symmetric homeomorphism between two complete metric spaces can be extended to a quasi-isometry between their hyperbolic approximations. This result is used to prove that two visual Gromov hyperbolic spaces are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundari…
In this paper we develop a complete theory of factorization for isometries of hyperbolic 4-space. Of special interest is the case where a pair of isometries is linked, that is, when a pair of isometries can be expressed each as compositions of two involutions, one of which is common to both isometries. Here we develop …
The study shows pseudo-Anosovs are common in mapping class groups.
problem Counting pseudo-Anosovs in mapping class groups.
method Using weakly contracting isometries and Morse elements.
result Pseudo-Anosovs are generic in mapping class groups.
Using the representation of the isometries as 2x2 invertible matrices over the division algebra $\H$ of quaternions, we give an algebraic characterization of the dynamical types of the orientation-preserving isometries of the hyperbolic 5-space. We also determine the conjugacy classes and the conjugacy classes of centr…
The paper studies quandles of hyperbolic 3-space isometries.
problem Investigating quandles of hyperbolic 3-space isometries.
method Introducing a new quandle and constructing a canonical map to the conjugate quandle.
result The canonical map from to the conjugate quandle is injective and has a discrete image.
Study on complex hyperbolic bidisk isometries and their Dirichlet domains.
problem Investigating isometries and Dirichlet domains in the complex hyperbolic bidisk.
method Examined the isometries of the complex hyperbolic bidisk and the Dirichlet domain formed by a cyclic subgroup action.
result Proved that the Dirichlet domain has two sides.
A bijection preserving horocycles/hypercycles is an isometry in hyperbolic plane.
problem Understanding transformations of constant curvature curves in hyperbolic geometry.
method Analyzing bijections that map horocycles to horocycles and hypercycles to hypercycles.
result Every abstract automorphism of geodesic/horocycles/hypercycles graphs is induced by an earthquake map/isometry.
Teichmüller space rigidity proven for Thurston metric.
problem Understanding isometries in Teichmüller space with Thurston metric.
method Analyzing -linear surjective isometries between cotangent spaces.
result Every isometry between hyperbolic surfaces induces an isometry in Teichmüller space.
Lattices in PSL(2,C) are omnipotent, acting on geodesics and homology.
problem Understanding the full isometry groups of hyperbolic 3-manifolds and their lattices.
method Analyzing geodesics and applying the Virtual Special Theorems.
result Every non-arithmetic lattice in PSL(2,C) is omnipotent, acting on homology.
L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete classification up to isometry, and also their isometry groups. The key tool is the so…
The study classifies horo-shrinkers in hyperbolic space under different isometries.
problem Characterizing horo-shrinkers in hyperbolic space under various isometries.
method Analyzing horo-shrinkers invariant by one-parameter groups of hyperbolic, parabolic, and spherical isometries.
result Grim reapers are defined as horo-shrinkers invariant by parabolic translations and are periodic surfaces.
We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all PU(,1) has involution length at most 8.
We investigate the rigidity of hyperbolic cone metrics on -manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…
New groups found with critical exponents close to but less than max.
problem Finding discrete isometry groups with critical exponents near maximum.
method Analyzing complex hyperbolic spaces to construct groups.
result Discrete isometry groups with critical exponents arbitrarily close to max but less.
We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any finite collection of finitely generated groups each of which either …
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic…
Extends Paulin's result to relatively hyperbolic groups.
problem Proving quasi-isometric equivalence between relatively hyperbolic groups.
method Introducing relative quasi-Mobius maps and using coarsely cusp-preserving quasi-isometries.
result Establishes a homeomorphism between Bowditch boundaries inducing quasi-Mobius maps.
This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space for n greater than 3. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and their contrast with the discrete groups of isometry of the hyperbolic 3-space.
This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.
Study describes moduli of quaternionic hyperbolic triples of points.
problem Tackles the congruence classes of triples of points in quaternionic hyperbolic space.
method Introduces invariants and defines quaternionic Goldman invariants for mixed configurations.
result Defines quaternionic analogues of Goldman invariants for mixed configurations.
In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions that closely resembles Minkowski spacetime. Viewing this in a modern context, we expand upon Macfarlane's idea and develop a model for real hyperbolic 3-space in which both points and isometries are expressed as complex quaternions, analogous…
The Jacobian of Douady-Earle extension equals 1 only for isometries.
problem Investigating the Jacobian of Douady-Earle extension maps.
method Analyzing the Jacobian of the Douady-Earle extension map and constructing sequences of hyperbolic surfaces.
result The Jacobian of the Douady-Earle extension map is 1 only when the map is an isometry, and it can grow arbitrarily large for certain sequences of surfaces.
Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not amenable then its second continuous bounded cohomology group with coefficients the regular representation does not vanish. This yields some structure results for such groups.
The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n > 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by Greenberg and Kojima, respectively. Our proof is non const…
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.
Rank One Hilbert Geometriesmath.GT
Develops Hilbert geometries and characterizes their isometries.
problem Characterizing isometries in Hilbert geometries.
method Defining rank one isometries and using geometric group theory.
result Discrete subgroups containing rank one isometries are either virtually cyclic or acylindrically hyperbolic.
Two groups with specific limit sets in hyperbolic spaces are identified.
problem Identifying convex cocompact subgroups with specific limit sets in real hyperbolic spaces.
method Examples of subgroups generated by reflections and rotations with limit sets as Pontryagin spheres and Menger curves.
result Examples of convex cocompact subgroups with limit sets as Pontryagin spheres and Menger curves are found.
The paper classifies discrete complex hyperbolic triangle groups.
problem Classifying discrete complex hyperbolic triangle groups.
method Analyzing complex hyperbolic spaces and isometries.
result Classifies discrete complex hyperbolic -triangle groups for .
The paper examines random walks on metric spaces and finds commensurable subgroups.
problem Determining commensurable subgroups via stationary measures in metric spaces.
method Analyzing random walks on isometry groups of metric spaces with non-singular stationary measures.
result Subgroups generated by random walks are commensurable under mild conditions.
Researchers express spectral determinants on hyperbolic cones.
problem Express spectral determinants on 2D hyperbolic cones.
method Explicitly expressed spectral determinants in terms of cone angle and geodesic radius.
result Results in recent paper by Freixas i Montplet and von Pippich were incorrect.
Complex hyperbolic triangle groups are discrete when certain conditions are met.
problem Proving discreteness of complex hyperbolic triangle groups.
method Analyzing representations and isometries to determine discreteness.
result Conditions for discreteness of complex hyperbolic triangle groups are identified.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
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…
This is an expository article about groups generated by two isometries of the complex hyperbolic plane.
Carrier graphs were first introduced for closed hyperbolic 3-manifolds by White. In this paper, we first generalize this definition to carrier graphs for representations of a rank two free group into the isometry group of hyperbolic three space. Then we prove the existence and the finiteness of minimal carrier graphs f…
New method classifies Heintze groups using -cohomology.
problem Quasi-isometry classification of Heintze groups.
method Introducing relative -cohomology and applying it to Heintze groups.
result Explicit construction of non-zero relative -cohomology classes.
Quasi-isometries in horospherical products are close to product maps.
problem Understanding the rigidity of quasi-isometries in specific geometric spaces.
method Proving quasi-isometries are uniformly close to product maps in horospherical products of hyperbolic spaces.
result Quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid.
Developed a half-space model for pseudo-hyperbolic space.
problem Modeling pseudo-hyperbolic space for any dimensions.
method Created an isometric embedding of pseudo-hyperbolic space into a half-space.
result Geodesics, totally geodesic submanifolds, horospheres, and isometry group are described in the half-space model.
The musical notes from a hyperbolic marimba can identify the shape of hyperbolic surfaces.
problem Identifying hyperbolic surfaces based on their musical notes.
method Assigning musical notes to geodesics hitting labeled curves on hyperbolic surfaces.
result The melodies produced by hyperbolic marimbas can characterize hyperbolic surfaces up to isometry.