A structure and representations of diffeomorphism groups of non-Archimedean manifolds

流形边界上的可控性与微分同胚群

Controllability and diffeomorphism groups on manifolds with boundary

本文研究具有光滑边界的流形的微分同胚群，证明其与边界微分同胚群构成具有局部截面的短正合序列，并建立向量场族的可控性结果，推广了Agrachev和Caponigro的结论，表明该群由指数映射的像生成。

Diffeomorphism Groups of Compact 4-manifolds are not always Jordan

We show that if $M$ is a compact smooth manifold diffeomorphic to the total space of an orientable $S^2$ bundle over the torus $T^2$, then its diffeomorphism group does not have the Jordan property, i.e., Diff$(M)$ contains a finite subgroup $G_n$ for any natural number $n$ such that every abelian subgroup of $G_n$ has index at leat $n$. This gives a counterexample to an old conjecture of Ghys.

一维流形微分同胚的几乎可约性、畸变与局部完美性

Almost reducibility, distortion and local perfection for diffeomorphisms of one-manifolds

本文证明圆周和实直线紧支撑光滑微分同胚群中元素畸变当且仅当几乎可约，并借助新的局部完美性结果建立该等价性。

光滑流形的初等等价与微分同胚群

Elementary equivalence and diffeomorphism groups of smooth manifolds

该研究证明了光滑流形的微分同胚群在初等等价下必然同胚，且在二维及以上维度中，保持光滑体积形式的微分同胚群也具有类似性质。

Anosov diffeomorphisms on Thurston geometric 4-manifolds

A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov diffeomorphisms.

The uniform perfectness of diffeomorphism groups of open manifolds

In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator length, those with support in balls and conjugation-generated norm.

Semidirect products of groups of loops and groups of diffeomorphisms of real, complex and quaternion manifolds

This article is devoted to the investigation of semidirect products of groups of loops and groups of diffeomorphisms of finite and infinte dimensional real, complex and quaternion manifolds. Necessary statements about quaternion manifolds with quaternion holomorphic transition mappings between charts of atlases are proved. Unitary representations of these groups including irreducible are constructed with the help of stochastic processes and quasi-invariant transition measures on groups $G$ relative to dense subgroups $G'$. It is proved, that this procedure provides a family of the cardinality …

Algebraic Structure of Diffeomorphism Groups of One-Manifolds

It is a celebrated result of Mather that the group of $C^k$--diffeomorphisms of an $n$--manifold is simple, provided that a mild isotopy condition is satisfied, with the possible exception of $k=n+1$. The purpose of this article is mostly expository, and in it we give a detailed account of Mather's proof in the case when $n=1$.

The Lie group of real analytic diffeomorphisms is not real analytic

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 construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analyt…

素三维流形的微分同胚群

Diffeomorphism groups of prime 3-manifolds

本文证明了基于Thurston几何$\text{Nil}$的紧致可定向非Haken三维流形的微分同胚群可形变收缩至等距群，从而完成了任意紧致可定向素三维流形微分同胚群同伦类型的确定。

Anosov diffeomorphisms of products II. Aspherical manifolds

We study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomorphisms. In the course of our study, we obtain a result of independent group theoretic and topological interest on the stability of the Hopf property, namely, that the product of finitely many Hopfian groups with trivial center is Hopfian.

Ricci flow and diffeomorphism groups of 3-manifolds

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $\text{Isom} (X,g)\to \text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.

Finite subgroups of diffeomorphism groups

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.

莫比乌斯带上保密度微分同胚群的拟态射

Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band

本文首次研究非可定向流形上保密度微分同胚群，证明莫比乌斯带上的该群存在可数多个线性无关的无界拟态射，并证明紧致连通带边非可定向曲面上的保密度微分同胚群弱可缩。