Graphically discrete groups have strong rigidity properties.
problem Understanding the rigidity of group actions on graphs.
method Introducing graphical discreteness and proving rigidity properties.
result Free products of graphically discrete groups are action rigid.
New infinite discrete group without finite quotients found.
problem Finding discrete groups without finite quotients.
method Constructing an infinite discrete subgroup of H3. result Infinite discrete subgroup with no finite non-trivial quotients.
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.
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.
Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
Factorizes discrete representations of finitely generated groups into PSL(2, R).
problem Understanding discrete representations of finitely generated groups into PSL(2, R).
method Factorization theorem for Fuchsian groups, Makanin-Razborov diagrams, and new class of groups called PSL(2, R)-discrete limit groups.
result Obtained useful information about PSL(2, R)-discrete limit groups.
Survey on Coxeter groups for Lie group examples.
problem Understanding Coxeter groups and their applications.
method Constructing discrete subgroups of Lie groups using Coxeter groups.
result Coxeter groups provide new examples in discrete subgroups of Lie groups.
Linking manifold structures to group dynamics.
problem Understanding geometric structures on manifolds and group dynamics.
method Study of geometric structures and representations of discrete groups into Lie groups.
result Recent findings on connections between manifold structures and group dynamics.
Study shows how metric space fundamental groups can be approximated by discrete groups.
problem Understanding fundamental groups of metric spaces through discrete approximations.
method Investigates the existence of isomorphisms between metric space fundamental groups and their discrete limits.
result Mild conditions are sufficient to establish an isomorphism between metric space fundamental groups and their discrete limits.
The study examines discrete subgroups of Lie groups and their residual finiteness.
problem Determining when discrete subgroups of Lie groups are residually finite.
method Analyzes known results and poses open questions.
result Answers to open questions will provide a comprehensive understanding of residual finiteness in Lie 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.
The paper finds discrete real specializations of braid group representations using Salem numbers.
problem Finding discrete real specializations of braid group representations.
method Using Salem numbers to find discrete real specializations of sesquilinear representations of braid groups.
result Details on the commensurability of target groups are provided.
Discrete connections on abelian Lie groups bundles are studied.
problem Understanding discrete connections on abelian Lie group principal bundles.
method Formalized discrete connections as singular cochains and proved a discrete holonomy formula.
result Discrete connections on abelian Lie group bundles have properties similar to continuous connections.
Discrete Lagrange problems solved with Lie group constraints.
problem Solving discrete Lagrange problems with Lie group constraints.
method Proving critical sections are solutions of unconstrained variational problems, applying Noether theory and multisymplectic forms.
result Critical sections of discrete Lagrange problems are solutions of unconstrained variational problems.
Survey on topological properties of discrete groups.
problem Understanding asymptotic topological properties of discrete groups.
method Exploring quasi-simple filtration, geometric simple connectivity, and topological inverse-representations.
result Discussing the conditions under which all finitely presented groups satisfy these properties.
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.
Universal inequalities for Laplacian eigenvalues on discrete groups.
problem Proving inequalities for Laplacian eigenvalues on discrete groups.
method Analyzing Laplacian eigenvalues with Dirichlet boundary conditions on subsets of discrete groups.
result Yang-type universal inequalities for Cayley graphs of amenable groups and the d-regular tree.
Discrete PU(1,1) representations of hyperelliptic groups are proven.
problem Characterizing PU(1,1) representations of hyperelliptic groups.
method Proving representations are basic if and only if they are discrete and faithful.
result A conjecture by S. Anan'in and E. Bento Gonçalves is partially proven.
The paper presents an algorithm to determine discreteness of certain groups.
problem Determining discreteness of specific groups generated by parabolic transformations.
method Algorithmic approach based on historical mathematical paradigms.
result Equivalence of different approaches to the discreteness problem.
This work extends reduction processes for nonholonomic discrete mechanical systems.
problem Nonholonomic discrete mechanical systems and their reductions.
method Introduces a category LDPd of discrete-time dynamical systems and a two-stage reduction process. result Two-stage reduction process produces systems isomorphic to one-stage reduction.
Determining the space of free discrete two generator groups of Möbius transformations is an old and difficult problem. In this paper we show how to construct large balls of full dimension in this space. To do this, we begin with a marked discrete group of non-separating disjoint circle type. Such a group determines thr…
Study shows critical exponents for tree-acting groups.
problem Understanding critical exponents of discrete groups on trees.
method Explicit construction of edge-indexed graphs.
result Proven existence of groups with specific critical exponents.
The study finds discrete subgroups with full limit sets in higher rank Lie groups.
problem Finding discrete subgroups with full limit sets in higher rank Lie groups.
method Analyzing real semi-simple Lie groups of higher rank and providing criteria for discrete subgroups of G=SL(3,R). result Existence of discrete subgroups with full limit sets in higher rank Lie groups.
New mechanics on non-associative octonions discovered.
problem Discrete mechanics on non-associative groups.
method Generalized Lagrangian and Hamiltonian mechanics to non-associative objects.
result Discrete mechanics on unitary octonions achieved.
Relates discrete group actions to orbit spaces as differentiable stacks.
problem Understanding dynamics of discrete groups on manifolds.
method Relating discrete group actions to orbit spaces as differentiable stacks.
result Orbit stack encodes dynamics up to conjugation and inversion.
Example found of subgroup not a lattice in product of Lie groups
problem Finding irreducible discrete subgroups that are not lattices
method Produced an example in SL(2,R)imesSL(2,R) result Example of subgroup not a lattice in product of Lie groups
We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…
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.
The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the modular group and, more generally, triangle groups. These are some of the simpl…
In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introdu…
We characterize finitely generated torsion-free Kleinian groups for which the real length spectrum (without multiplicities) is discrete.
This paper studies the probability of discrete groups generated by two parabolic elements in Kleinian groups.
problem Estimating the probability of discrete groups generated by two parabolic elements in Kleinian groups.
method Introduced geometrically natural probability measures and used computational investigation into the Riley slice.
result Identified the precise probability that a Fuchsian group generated by two parabolic Möbius transformations is discrete.
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invar…
Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
Irreducible groups cannot be free if they ergodically act on a boundary.
problem Characterizing discrete subgroups of Lie groups that act ergodically on boundaries.
method Analyzing the structure of discrete subgroups of real semi-simple Lie groups and their action on boundaries.
result Irreducible discrete subgroups of certain Lie groups cannot be free.
Improved homological dimension for certain subgroups in Lie groups.
problem Determining homological dimensions of discrete subgroups in Lie groups.
method Using recent results and properties of injectivity radius, the homological dimension gap is calculated.
result Infinite volume torsion-free subgroups of higher rank Lie groups have a homological dimension gap of at least 1/8 of the real rank.
Study polynomial trace identities in $SL(2,\IC)$ using quaternion algebras.
problem Understanding polynomial trace identities in $SL(2,\IC)$ and their applications.
method Use quaternion algebras over indefinites and their units to study discrete subgroups of $SL(2,\IC)$.
result Obtained structure theorems for quaternion algebras and new polynomial trace identities.
Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.
problem Characterizing dense subgroups of algebraic groups.
method Bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups.
result No C1-smooth slim limit set for higher rank semisimple algebraic groups. The paper proves that certain spaces have injective balls of any radius.
problem The injectivity radius of certain geometric spaces is infinite.
method Analyzes higher rank simple and semisimple Lie groups with specific properties.
result The locally symmetric spaces have injective balls of any radius.
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.
Discrete groups act properly on 3-space, solving Milnor's question.
problem Proper actions of discrete groups on 3-space.
method Historical review and recent progress.
result Existence of proper affine actions of free groups on 3-space.
Recent advances in discrete subgroups of SL2(C) influenced by Thurston's work.
problem The dynamics of discrete subgroups of SL2(C)
method Discussion of recent advances in the topic
result Influenced by Thurston's work, recent advances have been made in the dynamics of discrete subgroups of SL2(C)
We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their…
The study proves limitations on isospectral hyperbolic surfaces with discrete length spectra.
problem Characterizing isospectral hyperbolic surfaces with discrete length spectra.
method Utilizing Sunada's method and topological self-duplicating ends, the study explores isospectral families and their cardinality.
result Finite groups can be realized as full isometry groups of hyperbolic structures with discrete spectrum on surfaces with self-duplicating ends.
We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that C∞-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology o…
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…
Axis bundles in free-by-cyclic groups have non-generic monodromies.
problem Characterizing monodromies in free-by-cyclic groups.
method Analyzing monodromy actions on outer space and using properties of axis bundles.
result Monodromies having a 'lone axis' are non-generic and projectively discrete.