Smooth manifolds have equivalent diffeomorphism groups if and only if they are diffeomorphic.
problem Understanding when diffeomorphism groups of smooth manifolds are elementarily equivalent.
method Analyzing the equivalence of Cr and Cs diffeomorphism groups of smooth manifolds. result Equivalent diffeomorphism groups imply diffeomorphic manifolds, strengthening previous results.
Study on group cocycles for volume-preserving diffeomorphisms.
problem Understanding group cocycles on volume-preserving diffeomorphisms.
method Constructed two types of group cocycles on the volume-preserving diffeomorphism group.
result One cocycle yields the Euler class of flat sphere bundles for the sphere.
New Lie groups found for Poisson diffeomorphisms.
problem Finding Lie group structures on Poisson diffeomorphism groups.
method Using Poisson groupoids, develop Lie group structures.
result Poisson diffeomorphism groups of various Poisson manifolds are regular Lie groups.
Survey on foliations and diffeomorphism groups.
problem Relationship between algebraic and homotopical properties.
method Survey and analysis of existing literature.
result Explains the connection between diffeomorphism groups and foliations.
The paper explores the transitivity of orbifold diffeomorphisms.
problem Understanding the transitivity of orbifold diffeomorphisms.
method Investigates the group of compactly supported diffeomorphisms of orbifolds.
result The group of orbifold diffeomorphisms is n-transitive. Diffeomorphisms of convex polytopes form a Lie group.
problem Understanding transformations of convex polytopes.
method Forming a Lie group from diffeomorphisms of convex polytopes.
result The group of diffeomorphisms of a convex polytope is a regular Lie group.
Complex surfaces show non-simply connected diffeomorphism groups with non-homotopic loops.
problem Complex surfaces with non-simply connected diffeomorphism groups and non-homotopic loops.
method Exhibited examples of complex surfaces.
result Diffeomorphism groups of complex surfaces are not simply-connected and contain non-homotopic loops.
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.
The paper proves n-transitivity for equivariant diffeomorphisms of manifolds.
problem Proving n-transitivity for equivariant diffeomorphisms. method Analyzing the group of equivariant diffeomorphisms on proper smooth G-manifolds. result The group of equivariant diffeomorphisms acts n-transitively on M. Surface groups found in interval diffeomorphisms without global fix points.
problem Existence of surface groups in interval diffeomorphisms.
method Analyzing the structure of diffeomorphisms of the interval.
result Surface groups with specific properties in interval diffeomorphisms.
We determine the Riemannian manifolds for which the group of exact volume preserving diffeomorphisms is a totally geodesic subgroup of the group of volume preserving diffeomorphisms, considering right invariant L2-metrics. The same is done for the subgroup of Hamiltonian diffeomorphisms as a subgroup of the group of…
Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.
problem Understanding the structure of diffeomorphism groups of 3-manifolds.
method Deformation retraction and earlier work by multiple authors combined.
result Homotopy type of Diff(X) determined for prime 3-manifolds.
The paper studies quasimorphisms on density-preserving diffeomorphisms of the Möbius band.
problem Exploring quasimorphisms on groups of diffeomorphisms of non-orientable manifolds.
method Investigates the group of density-preserving diffeomorphisms on the Möbius band and shows the existence of unbounded quasimorphisms.
result The group of density-preserving diffeomorphisms on the Möbius band admits countably many unbounded quasimorphisms.
We prove that every right-angled Artin group embeds into the C∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the C∞ diffeomorphism group of the real line.
The paper examines uniform perfectness of diffeomorphism groups on open manifolds.
problem Uniform perfectness of diffeomorphism groups on open manifolds.
method Study of uniform perfectness, boundedness, and simplicity of diffeomorphism groups of compact and open manifolds.
result Obtained upper bounds of diameters for commutator length, balls, and conjugation-generated norm.
Generalizes π2-diffeomorphism finiteness to non-zero first homotopy groups.
problem Bounding diffeomorphic types of compact manifolds with vanishing first and second homotopy groups.
method Generalizing the π2-diffeomorphism finiteness theorem to include non-zero first homotopy groups. result Diffeomorphic types of compact manifolds with non-zero first homotopy groups can be bounded.
We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of diffeomorphism groups of compact smooth topological manifolds.
Study aspherical manifolds without Anosov diffeomorphisms.
problem Identify conditions under which certain manifolds do not support Anosov diffeomorphisms.
method Weakened conditions on Gogolev and Lafont, use group theoretic and topological results.
result Product of certain manifolds does not support Anosov diffeomorphisms.
Entropy norm on surface diffeomorphisms is unbounded.
problem Bounding entropy norm on surface diffeomorphisms.
method Analyzing the group of diffeomorphisms of a closed orientable surface.
result Entropy norm is unbounded on the group of diffeomorphisms of a closed orientable surface.
Study shows Hamiltonian diffeomorphisms form a connected component in C0-topology for most symplectic rational surfaces.
problem Understanding the C0-topology of symplectic diffeomorphisms on rational surfaces. method Combining techniques from symplectic mapping class groups and C0-symplectic topology, establishing C0-distance estimates. result Hamiltonian diffeomorphisms form a connected component in C0-topology for all but a few exceptions on rational surfaces. Affine diffeomorphisms are undistorted in half-translation surfaces.
problem Undistorted nature of affine diffeomorphism groups.
method Proof of undistorted subgroup property and systole map embedding.
result Finitely generated subgroups of affine diffeomorphisms are undistorted.
Conference compiles problems on foliations and diffeomorphisms.
problem Challenges in foliations and diffeomorphism groups.
method Compilation of problems from conference participants.
result Compilation of 20+ problems on foliations and diffeomorphisms.
Book introduces Hofer's metric on symplectic diffeomorphisms.
problem Understanding dynamics and growth in symplectic geometry.
method Introduces Hofer's metric and analyzes its properties.
result Provides insights into the structure of symplectic diffeomorphisms.
Positive paths connect diffeomorphisms on contact manifolds.
problem Defining and analyzing positivity in diffeomorphism groups of manifolds with contact structures.
method By examining paths of diffeomorphisms that are positively transverse to the contact distribution, showing flexibility and connecting diffeomorphisms.
result Any two diffeomorphisms on standard contact structure of R^(2n+1) can be connected by a positive path.
Study controllability of diffeomorphisms of simple polytopes.
problem Controllability of diffeomorphisms of simple polytopes.
method Lie group structure and controllability results for diffeomorphisms of simple polytopes.
result Identity component of the diffeomorphism group is generated by the exponential image.
We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the constructio…
Survey on realization problems for diffeomorphism groups.
problem Realizing subgroups of diffeomorphism groups.
method Discussion of recent results and open problems, including proofs of illustrative techniques.
result Illustration of key techniques in realization problems.
Proves constraints on groups extending Möbius transformations on spheres.
problem Constraints on groups extending Möbius transformations on spheres.
method Proved constraints through group transitivity and topological entropy analysis.
result Groups must be 4-transitive or arc 4-transitive, and contain elements of positive topological entropy.
Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.
problem Understanding the group of volume preserving diffeomorphisms through symplectic geometry.
method Using cotangent bundles of spaces of smooth embeddings, symplectic reduction, and nonlinear Grassmannians of augmented submanifolds.
result Descriptions of coadjoint orbits of the group of volume preserving diffeomorphisms in terms of submanifolds of augmented spaces.
We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω(ω-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω, up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extensio…
Finite groups of diffeomorphisms are uniquely determined by a vector field.
problem Identifying finite groups of diffeomorphisms using vector fields.
method Proving that every finite group of diffeomorphisms equals the centralizer of the group of smooth automorphisms of a G-invariant complete vector field. result Finite groups of diffeomorphisms are topologically determined by a vector field.
The study finds non-pseudoisotopic diffeomorphisms in certain 4-manifolds.
problem Identifying diffeomorphisms that are homotopic but not pseudoisotopic.
method Examples of diffeomorphisms in specific 4-manifolds.
result No non-pseudoisotopic diffeomorphisms in orientable 4-manifolds with free fundamental groups.
It is well-known that any isotopically connected diffeomorphism group G of a manifold determines uniquely a singular foliation $\F_G$. A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that…
In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that these groups contain the compactly supported diffeomorphisms. We finally show that …
In this paper we discuss the relationship between groups of diffeomorphisms of spheres and balls. We survey results of a topological nature and then address the relationship as abstract (discrete) groups. We prove that the identity component Diff_0(S^{2n-1}) of the group of smooth diffeomorphisms of S^{2n+1} admits no …
Exponential map proof for area-preserving diffeomorphisms on surfaces.
problem Proving the properties of the exponential map for a specific group of diffeomorphisms.
method Analyzing the Riemannian exponential map of the right-invariant L2 metric. result The exponential map is a nonlinear Fredholm map of index zero.
Study on controllability and groups of manifolds with boundaries.
problem Controllability of vector fields on manifolds with boundaries.
method Establish controllability results for diffeomorphism groups of manifolds with smooth boundaries.
result Diffeomorphism groups of manifolds with smooth boundaries form fibre bundles and are generated by the exponential map.
Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
problem Determining when cosymplectic manifolds are diffeomorphic based on their cosymplectomorphism groups.
method Characterized Reeb flow, used to descend isomorphism to symplectic base manifolds, preserved monodromy class ensuring bundle equivalence.
result Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
Survey of gauge theory for families of 4-manifolds.
problem Understanding gauge theory for families of 4-manifolds.
method Developed gauge theory for diffeomorphism groups of 4-manifolds.
result New applications of gauge theory to diffeomorphism groups.
Diffeomorphisms of surfaces have many unbounded quasi-morphisms.
problem Understanding the quasi-morphisms on surface diffeomorphism groups.
method Constructing a hyperbolic graph and using it to deduce properties of the groups.
result The group is not uniformly perfect and its fragmentation norm is unbounded.
Researchers simplify the computation of diffeomorphism groups for Morse-Bott foliations.
problem Computing the homotopy type of diffeomorphism groups for Morse-Bott foliations.
method Reduces the computation to three groups: diffeomorphisms of the critical manifold, vector bundle automorphisms, and fixed near the critical manifold.
result Shows how to compute the homotopy type of diffeomorphism groups for certain Morse-Bott foliations.
We study completeness properties of the Sobolev diffeomorphism groups Ds(M) endowed with strong right-invariant Riemannian metrics when the underlying manifold M is Rd or compact without boundary. The main result is that for s>dimM/2+1, the group Ds(M) is geodesically and me…
Some groups of real analytic diffeomorphism act n-transitively for each finite n.
We show that the group cohomology of the diffeomorphisms of the disk with n punctures has the cohomology of the braid group of n strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \…
This paper explains Mather's proof for simple groups of 1-manifold diffeomorphisms.
problem Understanding the simplicity of Ck--diffeomorphism groups of 1-manifolds. method Detailed exposition of Mather's proof for the case of 1-manifolds.
result Demonstrates the simplicity of Ck--diffeomorphism groups for k=2 in the case of 1-manifolds. For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1- manifolds. We show that all such actions…
Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.
problem Geometric analysis of fluid flows and optimal mass transport.
method Review of metrics and topology on diffeomorphism groups.
result Introduction of new metrics and topology for diffeomorphism groups.
We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant L2 or H1 metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the ri…