arXiv subjects

Isabelle Liousse

Publications and source records attributed to Isabelle Liousse.

15 recordsLinked to original sources

arXiv2026

Some elementary amenable subgroups of interval exchange transformations

In this paper, we study a family of finitely generated elementary amenable iet-groups. These groups are generated by finitely many rationals iets and rotations. For them, we state criteria for not virtual nilpotency or solvability, and we give conditions to ensure that they are not virtually solvable. We precise their abelianizations, we determine when they are isomorphic to certain lamplighter groups and we provide non isomorphic cases among them. As consequences, in the class of infinite finitely generated subgroups of iets up to isomorphism, we exhibit infinitely many non virtually solvable and non linear groups, and infinitely many solvable groups of arbitrary derived length.

math.GR↗

arXiv2023

Interval Exchange Transformations groups. Free actions and dynamics of virtually abelian groups

H\"older's theorem states that any group acting freely by circle homeomorphisms is abelian, this is no longer true for interval exchange transformations: we first give examples of free actions of non abelian groups. Then after noting that finitely generated groups acting freely by IET are virtually abelian, we classify the free actions of groups containing a copy of $\mathbb Z^2$, showing that they are ``conjugate" to actions in some specific subgroups $G_n$, namely $G_n \simeq ({\mathcal G}_2)^n \rtimes\mathcal S_n $ where ${\mathcal G}_2$ is the group of circular rotations seen as exchanges of $2$ intervals and $\mathcal S_n$ is the group of permutations of $\{1,...,n\}$ acting by permuting the copies of ${\mathcal G}_2$. We also study non free actions of virtually abelian groups and we obtain the same conclusion for any such group that contains a conjugate to a product of restricted rotations with disjoint supports and without periodic points. As a consequence, we provide examples of non virtually nilpotent subgroups of IETs. In particular, we show that the group generated by $f\in G_n$ periodic point free and $g\notin G_n$ is not virtually nilpotent. Moreover, we exhibit examples of finitely generated non virtually nilpotent subgroups of IETs, some of them are metabelian and others are not virtually solvable.

math.DS↗

arXiv2021

Uniform simplicity for subgroups of piecewise continuous bijections of the unit interval

Let $I=[0,1)$ and $\mathcal{PC}(I)$ [resp. $\mathcal{PC}^+(I)$] be the quotient group of the group of all piecewise continuous [resp. piecewise continuous and orientation preserving] bijections of $I$ by its normal subgroup consisting in elements with finite support (i.e. that are trivial except at possibly finitely many points). Unpublished Theorems of Arnoux ([Arn81b]) state that $\mathcal{PC}^+(I)$ and certain groups of interval exchanges are simple, their proofs are the purpose of the Appendix. Dealing with piecewise direct affine maps, we prove the simplicity of the group $\mathcal A^+(I)$ (see Definition 1.6). These results can be improved. Indeed, a group $G$ is uniformly simple if there exists a positive integer $N$ such that for any $f,\phi \in G\setminus\{Id\}$, the element $\phi$ can be written as a product of at most $N$ conjugates of $f$ or $f^{-1}$. We provide conditions which guarantee that a subgroup $G$ of $\mathcal{PC}(I)$ is uniformly simple. As Corollaries, we obtain that $\mathcal{PC}(I)$, $\mathcal{PC}^+(I)$, $PL^+ (\mathbb S^1)$, $\mathcal A(I)$, $\mathcal A^+(I)$ and some Thompson like groups included the Thompson group $T$ are uniformly simple.

math.GR↗

arXiv2019

Uniform perfectness for Interval Exchange Transformations with or without Flips

Let $\mathcal G$ be the group of all Interval Exchange Transformations. Results of Arnoux-Fathi ([Arn81b]), Sah ([Sah81]) and Vorobets ([Vor17]) state that $\mathcal G_0$ the subgroup of $\mathcal G$ generated by its commutators is simple. In [Arn81b], Arnoux proved that the group $\overline{\mathcal G}$ of all Interval Exchange Transformations with flips is simple. We establish that every element of $\overline{\mathcal G}$ has a commutator length not exceeding $6$. Moreover, we give conditions on $\mathcal G$ that guarantee that the commutator lengths of the elements of $\mathcal G_0$ are uniformly bounded, and in this case for any $g\in \mathcal G_0$ this length is at most $5$. As analogous arguments work for the involution length in $\overline{\mathcal G}$, we add an appendix whose purpose is to prove that every element of $\overline{\mathcal G}$ has an involution length not exceeding $12$.

math.GR↗

arXiv2019

Reversible Maps and Products of Involutions in Groups of IETS

An element $f$ of a group $G$ is reversible if it is conjugated in $G$ to its own inverse; when the conjugating map is an involution, $f$ is called strongly reversible. We describe reversible maps in certain groups of interval exchange transformations namely $G_n \simeq (\mathbb S^1)^n \rtimes\mathcal S_n $, where $\mathbb S^1$ is the circle and $\mathcal S_n $ is the group of permutations of $\{1,...,n\}$. We first characterize strongly reversible maps, then we show that reversible elements are strongly reversible. As a corollary, we obtain that composites of involutions in $G_n$ are product of at most four involutions. We prove that any reversible Interval Exchange Transformation (IET) is reversible by a finite order element and then it is the product of two periodic IETs. In the course of proving this statement, we classify the free actions of $BS(1,-1)$ by IET and we extend this classification to free actions of finitely generated torsion free groups containing a copy of $\mathbb Z^2$. We also give examples of faithful free actions of $BS(1,-1)$ and other groups containing reversible IETs. We show that periodic IETs are product of at most $2$ involutions. For IETs that are products of involutions, we show that such 3-IETs are periodic and then are product of at most $2$ involutions and we exhibit a family of non periodic 4-IETs for which we prove that this number is at least $3$ and at most $6$.

math.DS↗

arXiv2018

Nombre de classes de conjugaison d'\'el\'ements d'ordre fini dans les groupes de Brown-Thompson

We extend a result of Matucci on the number of conjugacy classes of finite order elements in the Thompson group $T$. According to Liousse, if $ gcd(m-1,q)$ is not a divisor of $r$ then there does not exist element of order $q$ in the Brown-Thompson group $T_{r,m}$. We show that if $ gcd(m-1,q)$ is a divisor of $r$ then there are exactly $\varphi(q). gcd(m-1,q)$ conjugacy classes of elements of order $q$ in $T_{r,m}$, where $\varphi$ is the Euler function phi. As a corollary, we obtain that the Thompson group $T$ is isomorphic to none of the groups $T_{r,m}$, for $m\not=2$ and any morphism from $T$ into $T_{r,m}$, with $m\not=2$ and $r\not= 0$ $mod \ (m-1)$, is trivial.

math.GR↗

arXiv2017

Distortion in groups of Affine Interval Exchange transformations

In this paper, we study distortion in the group $\mathcal A$ of Affine Interval Exchange Transformations (AIET). We prove that any distorted element $f$ of $\mathcal A$, has an iterate $f^ k$ that is conjugate by an element of $\mathcal A$ to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences we prove that no Baumslag-Solitar group, $BS(m,n)$ with $\vert m \vert \neq \vert n \vert$, acts faithfully by elements of $\mathcal A$, every finitely generated nilpotent group of $\mathcal A$ is virtually abelian and there is no distortion element in $\mathcal A_{\mathbb Q}$, the subgroup of $\mathcal A$ consisting of rational AIETs.

math.DS↗

arXiv2017

Any Baumslag-Solitar action on surfaces with a pseudo-Anosov element has a finite orbit

We consider $f, h$ homeomorphims generating a faithful $BS(1,n)$-action on a closed surface $S$, that is, $h f h^{-1} = f^n$, for some $ n\geq 2$. According to \cite{GL}, after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\Lambda$ of the action, included in $Fix(f)$. Here, we suppose that $f$ and $h$ are $C^1$ in neighbourhood of $\Lambda$ and any point $x\in\Lambda$ admits an $h$-unstable manifold $W^u(x)$. Using Bonatti's techniques, we prove that either there exists an integer $N$ such that $W^u(x)$ is included in $Fix(f^N)$ or there is a lower bound for the norm of the differential of $h$ only depending on $n$ and the Riemannian metric on $S$. Combining last statement with a result of \cite{AGX}, we show that any faithful action of $BS(1, n)$ on $S$ with $h$ a pseudo-Anosov homeomorphism has a finite orbit. As a consequence, there is no faithful $C^1$-action of $BS(1, n)$ on the torus with $h$ an Anosov.

math.DS↗

arXiv2014

Burnside problem for groups of homeomorphisms of compact surfaces

A group $Γ$ is said to be periodic if for any $g$ in $Γ$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and conjugate to a subgroup of $\mathrm{O}(3,\R)$ and we use it for proving that a finitely generated periodic group of spherical diffeomorphisms with even bounded orders is finite. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus (which is the purpose of a previous paper of the authors) is finite.

math.DS↗

arXiv2014

Burnside problem for measure preserving groups of toral homeomorphisms and for 2-groups of toral homeomorphisms

A group $G$ is said to be periodic if for any $g\in G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $μ$ is finite. Moreover if the group consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $\mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.

math.DS↗

arXiv2011

Actions of Baumslag-Solitar groups on surfaces

Let $BS(1,n) =< a, b \ | \ aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that BS(1,n) is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $. This paper deals with the dynamics of actions of BS(1,n) on closed orientable surfaces. We exhibit a smooth BS(1,n) action without finite orbits on $\TT ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid. We develop a general dynamical study for faithful topological BS(1,n)-actions on closed surfaces $S$. We prove that such actions $ $ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty. When $S= \TT^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of BS(1,n) on $\TT^2$. When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ then $fix(f)$ contains any minimal set.

math.DS↗

arXiv2010

$C^1$-actions of Baumslag-Solitar groups on $S^1$

Let $BS(1, n)=< a, b | aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that B(1, n) is isomorphic to the group generated by the two affine maps of the line : $f_0(x) = x + 1$ and $h_0(x) = nx $. The action on $S^1 = \RR \cup {\infty}$ generated by these two affine maps $f_0$ and $h_0 $ is called the standard affine one. We prove that any representation of BS(1,n) into $Diff^1(S^1)$ is (up to a finite index subgroup) semiconjugated to the standard affine action.

math.DS↗

arXiv2009

Rotation set and Entropy

In 1991 Llibre and MacKay proved that if $f$ is a 2-torus homeomorphism isotopic to identity and the rotation set of $f$ has a non empty interior then $f$ has positive topological entropy. Here, we give a converselike theorem. We show that the interior of the rotation set of a 2-torus $C^{1+ α}$ diffeomorphism isotopic to identity of positive topological entropy is not empty, under the additional hypotheses that $f$ is topologically transitive and irreducible. We also give examples that show that these hypotheses are necessary.

math.DS↗

arXiv2007

Singular measures of circle homeomorphisms with two break points

Let $T_{f}$ be a circle homeomorphism with two break points $a_{b},c_{b}$ and irrational rotation number $\varrho_{f}$. Suppose that the derivative $Df$ of its lift $f$ is absolutely continuous on every connected interval of the set $S^{1}\backslash\{a_{b},c_{b}\}$, that $DlogDf \in L^{1}$ and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\neq1$. We prove that the unique $T_{f}$- invariant probability measure $μ_{f}$ is then singular with respect to Lebesgue measure $l$ on $S^{1}$.

math.DS↗

arXiv2006

Rotation numbers in Thompson-Stein groups and applications

We study the properties of rotation numbers for some groups of piecewise linear homeomorphisms of the circle. We use these properties to obtain results on PL rigidity, non isomorphicity, non exoticity of automorphisms, non smoothability for Thompson-Stein groups.

math.DS↗