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.
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.
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.
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.
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.
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.
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.
In this paper we consider ultra-parallel complex hyperbolic triangle groups of type [m1,m2,0], i.e. groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra-parallel complex geodesics two of which intersect on the boundary. We prove some discreteness and non-discreteness…
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…
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.
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.
We construct an infinite discrete subgroup of the isometry group of H3 with no finite quotients other than the trivial group.
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…
We characterize finitely generated torsion-free Kleinian groups for which the real length spectrum (without multiplicities) is discrete.
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 describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics "at infinity" for representations of discrete groups into 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…
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.
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.
Complex Hyperbolic Triangle Groups of Type [m,m,0;3,3,2]math.GT In this paper we study discreteness of complex hyperbolic triangle groups of type [m,m,0;3,3,2], i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders 3,3,2 in complex geodesics with pairwise distances m,m,0. For fixed m, the parameter space of such groups is…
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.
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.
We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number δ between 0 and 21logq, there is a discrete subgroup Γ acting without inversion on a (q+1)-regular tree whose critical exponent is equal to δ. Explicit construction of edge-index…
A few years ago Kramer and Laubenbacher introduced a discrete notion of homotopy for simplicial complexes. In this paper, we compute the discrete fundamental group of the order complex of the Boolean lattice. As it turns out, it is equivalent to computing the discrete homotopy group of the 1-skeleton of the permutahedr…
We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.
The paper encourages Kleinian group thinking for higher rank Lie groups.
problem No specific problem stated; encouraging new thinking.
method Discussion of Kleinian group ideas applied to higher rank Lie groups.
result Encouragement to think about higher rank Lie groups using Kleinian group theory.
A new RG approach connects discrete and continuous time descriptions of Gaussian processes.
problem Discretization of continuous stochastic processes for accurate simulation or model inference.
method Renormalization Group (RG) approach for Gaussian time series generated by auto-regressive models.
result RG fixed points correspond to discretizations of linear SDEs, providing insights into process accuracy.
In this note we investigate to what extent the fundamental group of a metric space can be described as the inverse limit of its discrete fundamental groups. We show that some mild conditions suffice to imply the existence of an isomorphism and we provide a list of counterexamples to possible weakenings of these hypothe…