Groups of diffeomorphisms of one-manifolds, III: Nilpotent subgroups

On 3-manifolds that support partially hyperbolic diffeomorphisms

Let M be a closed 3-manifold that supports a partially hyperbolic diffeomorphism f. If $π_1(M)$ is nilpotent, the induced action of f on $H_1(M, R)$ is partially hyperbolic. If $π_1(M)$ is almost nilpotent or if $π_1(M)$ has subexponential growth, M is finitely covered by a circle bundle over the torus. If $π_1(M)$ is almost solvable, M is finitely covered by a torus bundle over the circle. Furthermore, there exist infinitely many hyperbolic 3-manifolds that do not support dynamically coherent partially hyperbolic diffeomorphisms; this list includes the Weeks manifold. If f is a strong partial…

Anosov diffeomorphisms on infra-nilmanifolds associated to graphs

Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, i.e. a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restr…

Fixed points for nilpotent actions on the plane and the Cartwright-Littlewood theorem

The goal of this paper is proving the existence and then localizing global fixed points for nilpotent groups generated by homeomorphisms of the plane satisfying a certain Lipschitz condition. The condition is inspired in a classical result of Bonatti for commuting diffeomorphisms of the 2-sphere and in particular it is satisfied by diffeomorphisms, not necessarily of class $C^{1}$, whose linear part at every point is uniformly close to the identity. In this same setting we prove a version of the Cartwright-Littlewood theorem, obtaining fixed points in any continuum preserved by a nilpotent act…

Groups of homeomorphisms of one-manifolds, III: Nilpotent subgroups

This paper has been withdrawn by the authors; it will be incorporated into part I of the series (in preparation).

与图相关的幂零李代数中Anosov有理形式的刻画

A characterization of Anosov rational forms in nilpotent Lie algebras associated to graphs

本文刻画了与图相关的幂零李代数中哪些有理形式对应的幂流形允许Anosov微分同胚，首次完整描述了一类具有多个非同构有理形式的所有Anosov情形，并纠正了文献中的错误结果。

A finitely generated, locally indicable group with no faithful action by C^1 diffeomorphisms of the interval

According to Thurston's stability theorem, every group of C^1 diffeomorphisms of the closed interval is locally indicable (.e., every finitely generated subgroup factors through Z). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the semi-direct product between F_2 an Z^2, although locally indicable, does not embed into Diff_+^1 (]0,1[). (Here F_2 is any free subgroup of SL(2,Z), and its action on Z^2 is the projective one.) Moreover, we show that for every non-solvable subgroup G of SL(2,Z), the semi-direct product bet…

Recurrent orbits of subgroups of local complex analytic diffeomorphisms

We show recurrent phenomena for orbits of groups of local complex analytic diffeomorphisms that have a certain subgroup or image by a morphism of groups that is non-virtually solvable. In particular we prove that a non-virtually solvable subgroup of local biholomorphisms has always recurrent orbits, i.e. there exists an orbit contained in its set of limit points.

Somewhere dense orbit of abelian subgroup of diffeomorphisms maps acting on C^n

In this paper, we give a characterization for any abelian subgroup G of a lie group of diffeomorphisms maps of C^n, having a somewhere dense orbit G(x), x in C^n: G(x) is somewhere dense in C^n if and only if there are f_{1},....,f_{2n+1 in exp^{-1}(G) such that f_{2n+1} in vect(f_{1},...,f_{2n}) and Z.f_{1}(x)+....+Z.f_{2n+1}(x) is dense subgroup of C^n, where vect(f_{1},....,f_{2n}) is the vector space over R generated by f_{1},....,f_{2n}.

Equidistribution of Dense Subgroups on Nilpotent Lie Groups

Let $Γ$ be a dense subgroup of a simply connected nilpotent Lie group $G$ generated by a finite symmetric set $S$. We consider the $n$-ball $S_n$ for the word metric induced by $S$ on $Γ$. We show that $S_n$ (with uniform measure) becomes equidistributed on $G$ with respect to the Haar measure as n tends to infinity. We give rates and also prove the analogous result for random walk averages (i.e. the local limit theorem).

Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of $S^2$

We show that if $M$ is a compact oriented surface of genus 0 and $G$ is a subgroup of $\Symp^ω_μ(M)$ which has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer of an infinite order $f \in \Symp^ω_μ(M)$ is virtually abelian. Another immediate corollary is that if $G$ is a solvable subgroup of $\Symp^ω_μ(M)$ then $G$ is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of $\Symp^ω_μ(S^2).$

Homomorphisms between diffeomorphism groups

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 have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if M is any closed manifold, and Diff(M)_0 injects into the diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the answ…

Finite orbits for nilpotent actions on the torus

A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.

Sur les sous-groupes nilpotents du groupe de Cremona

We describe the nilpotent subgroups of the group Bir(P^2(C)) of birational transformations of the complex projective plane. Let N be a nilpotent subgroup of class k>1; then either each element of N has finite order, or N is virtually metabelian.

Reducing conjugacy in the full diffeomorphism group of R to conjugacy in the subgroup of orientation-preserving maps

Let $\Diffeo=\Diffeo(\R)$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\R$, under the operation of composition, and let $\Diffeo^+$ be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements $f,g\in \Diffeo$ are conjugate in $\Diffeo$ to associated conjugacy problems in the subgroup $\Diffeo^+$. The main result concerns the case when $f$ and $g$ have degree -1, and specifies (in an explicit and verifiable way) precisely what must be added to t…

Growth of groups and diffeomorphisms of the interval

We prove that the so called Grigorchuk-Maki group of intermadiate growth can be seen as a group of $C^1$ diffeomorphisms of the interval. On the other hand, we prove that every group of $C^{1+α}$ diffeomorphisms of the interval having subexponential growth is almost nilpotent.

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.

On the sharp regularity for arbitrary actions of nilpotent groups on the interval: the case of N_4

We determine the largest α for which the nilpotent group of 4-by-4 triangular matrices with integer coefficients and 1 in the diagonal embeds into the group of C1+α diffeomorphism of the closed interval.

Derived length of solvable groups of local diffeomorphisms

Let $G$ be a solvable subgroup of the group $\diff{}{n}$ of local complex analytic diffeomorphisms. Analogously as for groups of matrices we bound the solvable length of $G$ by a function of $n$. Moreover we provide the best possible bounds for connected, unipotent and nilpotent groups.

Harmonic measures on Bowditch boundaries of groups hyperbolic relative to virtually nilpotent subgroups

For a group hyperbolic relative to virtually nilpotent subgroups, on a cusped graph associated to the group, we construct a random walk whose Martin boundary is the Bowditch boundary of the group. Moreover, the harmonic measure is a conformal density corresponding to a hyperbolic Green metric and is exact dimensional on the Bowditch boundary. The latter equipped with a visual distance induced by the Green metric is an Ahlfors-regular metric measure space. The dimension is given in terms of the drift, a Green drift and the asymptotic entropy. The Patterson-Sullivan density for the action on the…

Groups of germs of analytic diffeomorphisms in (C^2,0)

Let G a group of germs of analytic diffeomorphisms in (C^2,0). We find some remarkable properties supposing that G is finite, linearizable, abelian nilpotent and solvable. In particular, if the group is abelian and has a generic dicritic diffeomorphisms, then the group is a subgroup of a 1-parametric group. In addition, we study the topological behaviour of the orbits of a dicritic diffeomorphisms. Last, we find some invariants in order to know when two diffeomorphisms are formally conjugates.

Groups of smooth diffeomorphisms of Cantor sets embedded in a line

Let K be a Cantor set embedded in the real line R. Following Funar and Neretin, we define the diffeomorphism group of K as the group of homeomorphisms of K which locally look like a diffeomorphism between two intervals of R. Higman-Thompson's groups Vn appear as subgroups of such groups. In this article, we prove some properties of this group. First, we study the Burnside problem in this group and we prove that any finitely generated subgroup consisting of finite order elements is finite. This property was already proved by Rover in the case of the groups Vn. We also prove that any finitely ge…

Finite dimensional groups of local diffeomorphisms

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic group up to add extra formal diffeomorphisms. We show that this is the case for virtually polycyclic subgroups and in particular finitely generated virtually nilpotent groups of local biholomorphisms. We provide several methods to identify this property and build examples. As a consequence we generalize results of Arnold, Seigal-Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, …

nilpotent群作用于一维紧致流形的临界正则性

Critical regularity of nilpotent groups acting on one-dimensional compact manifolds

研究nilpotent群在闭区间或半开区间及圆上的忠实微分作用的临界正则性，通过代数方法推导出表达式，推广了先前成果。

Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic group of finite index or a nonabelian free subgroup.