Complex hyperbolic isometries are decomposed into involutions, revealing specific lengths.
problem Understanding the structure of complex hyperbolic isometries.
method Analyzing decompositions of complex hyperbolic isometries into products of involutions.
result PU(2,1) has involution length 4 and commutator length 1, and PU(n,1) has involution length at most 8 for all n ≥ 3.
Let Hn denote the complex hyperbolic space of dimension n. The group U(n,1) acts as the group of isometries of Hn. In this paper we investigate when two isometries of the complex hyperbolic space commute. Along the way we determine the centralizers.
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…
Study of special elliptic isometries and their lengths in complex hyperbolic plane.
problem Classifying lengths of special elliptic isometries in complex hyperbolic plane.
method Classification and description of relative SU(2,1)-character varieties.
result Fully classified lengths of special elliptic isometries (2, 3, 4).
Characterizes contracting isometries in CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups.
problem Characterizing contracting isometries in CAT(0) cube complexes and their relation to acylindrical hyperbolicity.
method Characterization of contracting isometries without local finiteness assumption, combinatorial boundary introduction, and application to diagram groups.
result Determine precise conditions for acylindrical hyperbolicity of diagram groups.
A new model uses complex quaternions for hyperbolic 3-space.
problem No specific problem stated; focuses on a new model.
method Developed a model using complex quaternions for real hyperbolic 3-space.
result Introduced new computational tools for studying hyperbolic isometries.
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.
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.
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.
Complex Hyperbolic Triangle Groups of Type [m,m,0;3,3,2]math.GT Study on discrete complex hyperbolic triangle groups of specific type.
problem Discreteness of complex hyperbolic triangle groups of type [m,m,0;3,3,2]. method Analysis of groups generated by complex reflections with specific orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
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 α-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.
This paper calculates hyperbolicity constants for pants and relative pants graphs.
problem Understanding how hyperbolicity constants change based on the surface.
method Study of hyperbolicity constants for pants and relative pants graphs for specific surfaces.
result Calculates hyperbolicity constants for the five-punctured sphere and twice punctured torus.
New representations of 3-manifold groups into complex hyperbolic space found.
problem Finding representations of 3-manifold groups into complex hyperbolic spaces.
method Using Lefschetz fibrations and orbifold fundamental groups of branched coverings of the projective plane.
result Infinitely many non-conjugate representations discovered.
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 (n,∞,∞)-triangle groups for n=3,4,5. Lectures on complex hyperbolic spaces and their groups.
problem Understanding interactions between complex hyperbolic spaces and discrete groups.
method Discussion of function theory and discrete group theory.
result Interactions between complex hyperbolic spaces and discrete groups.
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.
Complex Hyperbolic Triangle Groups of Type [m,m,0;n1,n2,2]math.GT The study determines discreteness of complex hyperbolic triangle groups.
problem Discreteness of complex hyperbolic triangle groups of specific type.
method Analysis of groups generated by complex reflections with given orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
This is an expository article about groups generated by two isometries of the complex hyperbolic plane.
CAT(0) cube complexes have special isometry groups.
problem Characterizing isometry groups of CAT(0) cube complexes.
method Analyzing subcomplexes and using hyperbolicity properties.
result If Aut(X) ≠ Isom(X), X has a product structure.
New condition prevents hyperbolic spaces from matching curve complexes.
problem Identifying when hyperbolic spaces cannot match curve complexes.
method Analyzing specific hyperbolic complexes and identifying a condition.
result Identified a condition preventing quasi-isometry between hyperbolic spaces and curve complexes.
Classifies CR submanifolds in complex hyperbolic spaces.
problem Understanding CR submanifolds in complex hyperbolic spaces.
method Classifying orbits of a subgroup of the solvable part of the Iwasawa decomposition.
result Classification of homogeneous CR submanifolds.
Every Gromov hyperbolic group can be described by a finite subdivision rule.
problem Describing Gromov hyperbolic groups using subdivision rules.
method Finite subdivision rules acting on the 3-sphere to describe groups.
result Extends the representation of hyperbolic groups to non-cubulated groups.
In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.
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.
A k-reflection of the n-dimensional complex hyperbolic space ${\rm H}_{\C}^n$ is an element in U(n,1) with negative type eigenvalue λ, ∣λ∣=1, of multiplicity k+1 and positive type eigenvalue 1 of multiplicity n−k. We prove that a holomorphic isometry of ${\rm H}_{\C}^n$ is a product of at most fou…
Researchers prove unique bisectors in a complex hyperbolic space.
problem Non-uniqueness of bisectors in symmetric spaces.
method Analyzed bisectors in the bidisk, a rank 2 geometry.
result Bisectors uniquely determine a pair of points in H2imesH2. Complex hyperbolic Kleinian groups yield Stein manifolds under certain conditions.
problem Characterizing discrete groups acting on complex hyperbolic spaces.
method Proving conditions for a discrete group to yield a Stein manifold.
result If a discrete group is convex-cocompact, torsion-free, and has a critical exponent less than 2, the quotient manifold is Stein.
This paper classifies minimal complexity hyperbolic 3-manifolds with geodesic boundaries.
problem Classifying minimal complexity hyperbolic 3-manifolds with geodesic boundaries.
method Defined and studied the class Mg,k of smallest complexity manifolds with k torus cusps and connected totally geodesic boundary of genus g. result Provided a complete classification of manifolds in Mk,k and Mk+1,k, describing their isometry groups and commensurability invariants. Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.
problem Structure of minimal displacement set in weakly systolic complexes.
method Investigation of minimal displacement set properties and embeddings.
result Minimal displacement set is systolic and embeds isometrically into the complex.
We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodes…
Study extends biholomorphisms between convex domains in complex space without boundary constraints.
problem Extending biholomorphisms between convex domains without boundary regularity.
method Combining coarse geometry techniques with dynamical properties of maps in Gromov hyperbolic spaces.
result Proves extensions for biholomorphisms and quasi-isometries between convex domains.
The study resolves conjectures about quasiflats in hierarchically hyperbolic spaces.
problem Understanding the structure and properties of quasiflats in hierarchically hyperbolic spaces.
method Proving that any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants.
result The rank of a hierarchically hyperbolic space coincides with the maximal dimension of a quasiflat.
We equip the whole tangent space TM to a hyperbolic manifold M (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of M extend to isometries of TM by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…
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.
We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with …
Classifies matrices in the quaternionic hyperbolic unitary group.
problem Understanding the structure of matrices in the quaternionic hyperbolic unitary group.
method Used complex representation and characteristic polynomial to study matrices.
result Computed the characteristic polynomial and studied its sign.
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.
The paper introduces new invariants to detect hyperbolic parts in relatively hyperbolic groups.
problem Detecting hyperbolic parts in relatively hyperbolic groups.
method Quasi-isometry invariants based on small cancellation theory over free products.
result Infinitely many quasi-isometry types of one-ended hyperbolic relative groups can be constructed.
We extend a theorem of Masur and Wolf which says that given a hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four-holed sphere, the one-holed torus, a…
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.
New lattice extensions of Schottky groups in hyperbolic space.
problem Understanding complex translation lengths in hyperbolic manifolds.
method Produced systolic lattice extensions of Schottky subgroups.
result Density of complex translation lengths in closed hyperbolic manifolds.
This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by…
The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.
problem Determining a closed convex set in hyperbolic 3-space by its boundary metric.
method Pogorelov's rigidity theorem, Hausdorff measure, and complex analysis techniques.
result The intrinsic path metric on the boundary determines a closed convex set up to isometry under certain conditions.
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.