The abstract discusses various open problems in mapping class groups.
problem Open problems in mapping class groups of surfaces.
method Discussion of open problems.
result Various open problems in mapping class groups.
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We begin with a review of the notion of a braid group. We then discuss some known solutions to decision problems in braid groups. We then move on to proving new results in braid group algorithmics. We offer a quick solution to the generalized word problem in braid groups, in the special case of cyclic subgroups. We ill…
Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.
problem Characterizing and understanding group actions on surfaces, especially maximal handlebody and Hurwitz groups.
method Analyzing various group actions, comparing Hurwitz and handlebody groups, and examining bounding actions.
result Relationship between Hurwitz groups and maximal handlebody groups, and insights into geometric bounding actions.
Research classifies geometric structures on manifolds using surface group representations.
problem Classifying geometric structures on manifolds related to surface group actions on character varieties.
method Surveying results on surface groups, focusing on compact and non-compact target groups, discussing various representations and their dynamics.
result Dichotomy in dynamics of character varieties based on compactness of target groups.
Predicts whether online discussions will be productive or not.
problem Identifying wasteful discussions in group settings.
method Analyze conversational dynamics and linguistic cues.
result Linguistic cues and conversational patterns predict discussion productivity.
Paper discusses groups where twisted Alexander polynomials vanish.
problem Understanding groups with vanishing twisted Alexander polynomials.
method Introduced and studied twisted Alexander vanishing groups, constructed knots, and analyzed representations.
result Every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.
In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense sub…
The study explores normal generators for mapping class groups and their properties.
problem Understanding normal generators for mapping class groups of surfaces.
method Examined the relation between normal generation and asymptotic translation lengths on Teichmüller space and curve graph.
result Discussed several open questions related to normal generators.
We discuss which Kleinian groups are commensurable with Kleinian groups generated by rotations, with particular emphasis on Kleinian groups that arise from Dehn surgery on a knot.
This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous negatively curved manifolds. A main conjecture provides a general description; an extend…
In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating combinatorial descriptions and geometric properties of hyperbolic 3-manifolds.
We discuss in this article a property of action of groups by isometries called "well displacing". An action is said to be well displacing, if the displacement function is equivalent to the the displacement function for the action on the Cayley graph. We relate this property with the fact that orbit maps are quasi-isome…
We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations.
In this survey, we discuss a series of linearization problems--for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of averaging leads to problems concerning almost actions of compact groups and alm…
Gerbes on quantum groupsmath.DG
We discuss an approach to quantum gerbes over quantum groups in terms of q-deformation of transition functions for a loop group bundle. The case of the quantum group SUq(2) is treated in some detail.
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The f…
The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.
problem Reconstructing planar curves with specified Euclidean or affine curvatures.
method The paper presents algorithms for curve reconstruction under the special Euclidean and equi-affine groups.
result The reconstructed curves are close to the original curves in terms of the specified curvatures.
The study characterizes geodesic orbit Riemannian spaces and their properties.
problem Characterizing geodesic orbit Riemannian spaces and their properties.
method Analyzing the structure of nilradical and radical of Lie algebra, discussing compact Lie group representations.
result Described the structure of nilradical and radical of Lie algebra of isometry group.
Constructs 2-representations and 2-vector bundles for Lie 2-groups.
problem No strict model for string 2-group, so coherent model is constructed.
method Builds category of 2-representations and equivariant 2-vector bundles.
result Explicit formulas for 2-representations and 2-vector bundles.
Book on infinite-dimensional Lie groups, covering basics and various classes.
problem Understanding Lie groups in infinite-dimensional spaces.
method Develops smooth manifolds and Lie groups in locally convex spaces, discussing various classes.
result Detailed exploration of infinite-dimensional Lie groups and their properties.
The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.
problem Conjectures about the reciprocity of Chern-Simons invariants of 3-manifolds.
method Supporting evidence through Galois descent of a -group.
result The conjectures hold under the condition of Galois descent of a -group.
Hyperreflection groupsmath.GR
We introduce the concept of hyperreflection groups, which are a generalization of Coxeter groups. We prove the Deletion and Exchange Conditions for hyperreflection groups, and we discuss special subgroups and fundamental sectors of hyperreflection groups. In the second half of the paper, we prove that Coxeter groups an…
This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters).
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.
Survey on RAAGs in low-dimensional manifold diffeomorphisms.
problem Role of RAAGs in diffeomorphism groups of low-dimensional manifolds.
method Subgroup structure, algebraic structure, compactness, regularity analysis.
result RAAGs have diverse actions on low-dimensional manifolds, with restrictions on higher regularity.
This is partly a survey and partly a research article. Some known results and open problems about Kaehler groups (fundamental groups of compact Kaehler manifolds) are discussed. A new notion of Kaehler homomorphism is introduced. This is a homomorphism induced by a holomorphic map between these types of manifolds. Some…
A group is said to be bounded if it has a finite diameter with respect to any bi-invariant metric. In the present paper we discuss boundedness of various groups of diffeomorphisms.
New cohomology groups generalize Euler number for Lie superalgebras.
problem Generalizing cohomology groups for Lie superalgebras.
method Abstracted Poisson cohomology groups to Poisson-like cohomology groups for general Lie superalgebras.
result De Rham cohomology groups match Poisson-like cohomology groups for differential forms.
We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space o…
It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
The history and proofs of Topological Tverberg Theorem are discussed.
problem Understanding the proofs and properties of Topological Tverberg Theorem.
method Historical overview and detailed cell structure analysis of classifying space.
result Clarification of the cell structure of the classifying space .
Survey article on loop groups and their representations, following a course of three lectures held at the summer school "algebraic groups" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations.
Stability of Lie group homomorphisms and subgroups via Moser type argument.
problem When a deformation of Lie group homomorphisms and subgroups is trivial.
method Moser type argument for compact groups.
result Stability results for compact Lie groups.
Maps configuration spaces to groups for large k.
problem Understanding connections between configuration spaces and groups .
method Constructs a map from fundamental groups of complements to .
result Discovers connections between different groups.
Notes on Chamanara's surface with a non-elementary Fuchsian group.
problem Characterizing the Veech group of a specific translation surface.
method Review and analysis of Chamanara's proof.
result The Veech group is a non-elementary Fuchsian group generated by two parabolic elements.
Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
Knot Group Epimorphismsmath.GT
Any knot group is the image of the group of a prime knot by a homomorphism that preserves peripheral structure. In fact, there are infinitely many such prime knots. A related partial order on knots is defined, and its properties are discussed.
A free action of a finite group on an odd-dimensional sphere is said to be almost linear if the action restricted to each cyclic or 2-hyperelementary subgroup is conjugate to a free linear action. We begin this survey paper by reviewing the status of almost linear actions on the 3-sphere. We then discuss almost linear …
Defines ternary group homology for knot theory applications.
problem Understanding ternary groups and their homology.
method Developed a homology theory for ternary groups using associativity and skew elements.
result Discussed applications of ternary knot groups.
We discuss a general procedure for using characteristic classes to study the components of the gauge group for a principal G-bundle. To illustrate this, we work out the case where G is the projective unitary group.
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…
Loop group method varies with base point choice.
problem Dependence of loop group method on base point choice.
method Analyzes how loop group method for harmonic maps varies with base point.
result Loop group method results depend on base point selection.
We prove a squeezing/stability theorem for delta-epsilon controlled L-groups when the control map is a fibration on a finite polyhedron. A relation with boundedly-controlled L-groups is also discussed.
The abstract discusses ribbon graphs and their relation to surface groups.
problem Understanding the relationship between ribbon graphs and surface groups.
method Formal exposition based on Labourie's notes.
result Formalizes the relationship between ribbon graphs and surface groups.
We discuss a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov--Zabolotskaya equation and the Boyer--Finley equation.
In this paper we mainly pay attention to the complex hyperbolic triangle groups of type (m, n, infinity) and discuss the discreteness. From the results more explicit conclusions about the triangle groups of type (n, infinity, infinity) will also be given.
Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.
The abstract discusses extensions of Jacobi groups and their orbit space properties.
problem Understanding the properties of extended Jacobi groups and their orbit spaces.
method Proving an analogue of Chevalley Theorem and constructing a Frobenius structure.
result Construction of a Dubrovin Frobenius structure on the orbit space.