Piecewise Linear Homeomorphisms Research Articles

Non-smoothability for a class of groups of piecewise linear homeomorphisms of the interval

Non-smoothability for a class of groups of piecewise linear homeomorphisms of the interval

A combinatorial criterion and center for the quasi-isometry groups of Euclidean spaces

In this study, we introduce the notion of PLδ-homeomorphisms of Rn. Furthermore, we provide a combinatorial criterion reliant on the vertices and edges of simplicial structures, to determine whether a piecewise-linear homeomorphism to be a quasi-isometry. By employing this criterion, we subsequently show that the center of the group QI(Rn), which comprises all quasi-isometries of Rn, is indeed trivial.

A complete characterization of reversibility in PL(S)

An element g in a group G is called reversible if g−1=hgh−1 for some h∈G. We give a complete characterization of reversibility in the group PL(S) of piecewise linear homeomorphisms of the circle with finite number of break points. This work is the continuation of the study done by K. Ben Rejeb and H. Marzougui in (2019) [2], in particular we answer a question raised in it. This work is also a comparison and an extension to results of reversibility in the groups PL(R) and Homeo(S); some interesting examples are constructed here to answer questions that can be raised in this subject.

The BNSR-invariants of the Lodha–Moore groups, and an exotic simple group of type

AbstractIn this paper we give a complete description of the Bieri–Neumann–Strebel–Renz invariants of the Lodha–Moore groups. The second author previously computed the first two invariants, and here we show that all the higher invariants coincide with the second one, which finishes the complete computation. As a consequence, we present a complete picture of the finiteness properties of normal subgroups of the first Lodha–Moore group. In particular, we show that every finitely presented normal subgroup of the group is of type $\textrm{F}_\infty$ , answering a question posed in Oberwolfach in 2018. The proof involves applying a variation of Bestvina–Brady discrete Morse theory to the so called cluster complex X introduced by the first author. As an application, we also demonstrate that a certain simple group S previously constructed by the first author is of type $\textrm{F}_\infty$ . This provides the first example of a type $\textrm{F}_\infty$ simple group that acts faithfully on the circle by homeomorphisms, but does not admit any nontrivial action by $C^1$ -diffeomorphisms, nor by piecewise linear homeomorphisms, on any 1-manifold.

Direct decompositions of groups of piecewise linear homeomorphisms of the unit interval

We study subgroups of the group [Formula: see text] of piecewise linear orientation-preserving homeomorphisms of the unit interval [Formula: see text] that are differentiable everywhere except at finitely many real numbers, under the operation of composition. We provide a criterion for any two subgroups of [Formula: see text] which are direct products of finitely many indecomposable non-commutative groups to be non-isomorphic. As its application, we give a necessary and sufficient condition for any two subgroups of the R. Thompson group [Formula: see text] that are stabilizers of finite sets of numbers in the interval [Formula: see text] to be isomorphic, thus solving a problem by G. Golan and M. Sapir. We also show that if two stabilizers are isomorphic, then they are conjugate inside a certain group [Formula: see text].

Determining solubility for finitely generated groups of PL homeomorphisms

The set of finitely generated subgroups of the group P L + ( I ) PL_+(I) of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson’s group F F . Here, we show that every finitely generated subgroup G > P L + ( I ) G>PL_+(I) is either soluble, or contains an embedded copy of the finitely generated, non-soluble Brin-Navas group B B , affirming a conjecture of the first author from 2009. In the case that G G is soluble, we show the derived length of G G is bounded above by the number of breakpoints of any finite set of generators. We specify a set of ‘computable’ subgroups of P L + ( I ) PL_+(I) (which includes R. Thompson’s group F F ) and give an algorithm which determines whether or not a given finite subset X X of such a computable group generates a soluble group. When the group is soluble, the algorithm also determines the derived length of ⟨ X ⟩ \langle X\rangle . Finally, we give a solution of the membership problem for a particular family of finitely generated soluble subgroups of any computable subgroup of P L + ( I ) PL_+(I) .

The BNSR-invariants of the Stein group 𝐹2,3

Abstract The Stein group F 2 , 3 F_{2,3} is the group of orientation-preserving piecewise linear homeomorphisms of the unit interval with slopes of the form 2 p ⁢ 3 q 2^{p}3^{q} ( p , q ∈ Z p,q\in\mathbb{Z} ) and breakpoints in Z ⁢ [ 1 6 ] \mathbb{Z}[\frac{1}{6}] . This is a natural relative of Thompson’s group 𝐹. In this paper, we compute the Bieri–Neumann–Strebel–Renz (BNSR) invariants Σ m ⁢ ( F 2 , 3 ) \Sigma^{m}(F_{2,3}) of the Stein group for all m ∈ N m\in\mathbb{N} . A consequence of our computation is that (as with 𝐹) every finitely presented normal subgroup of F 2 , 3 F_{2,3} is of type F ∞ \operatorname{F}_{\infty} . Another, more surprising, consequence is that (unlike 𝐹) the kernel of any map F 2 , 3 → Z F_{2,3}\to\mathbb{Z} is of type F ∞ \operatorname{F}_{\infty} , even though there exist maps F 2 , 3 → Z 2 F_{2,3}\to\mathbb{Z}^{2} whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in Σ ∞ ⁢ ( F 2 , 3 ) \Sigma^{\infty}(F_{2,3}) , but there exist (non-discrete) characters that do not even lie in Σ 1 ⁢ ( F 2 , 3 ) \Sigma^{1}(F_{2,3}) . To the best of our knowledge, F 2 , 3 F_{2,3} is the first group whose BNSR-invariants are known exhibiting these properties.

Parrondo’s Paradox for Tent Maps

In this paper, we study the dynamic Parrondo’s paradox for the well-known family of tent maps. We prove that this paradox is impossible when we consider piecewise linear maps with constant slope. In addition, we analyze the paradox “simple + simple = complex” when a tent map with constant slope and a piecewise linear homeomorphism with two different slopes are considered.

Matrix group actions on product of spheres

Let [Formula: see text] be the special linear group over integers and [Formula: see text] [Formula: see text], or [Formula: see text] products of spheres and tori. We prove that any group action of [Formula: see text] on [Formula: see text] by diffeomorphims or piecewise linear homeomorphisms is trivial if [Formula: see text] This confirms a conjecture on Zimmer’s program for these manifolds.

Invariable generation of certain groups of piecewise linear homeomorphisms of the interval

Let P P be the group of all the orientation preserving piecewise linear homeomorphisms of the interval [ 0 , 1 ] [0,1] . Given any a > 1 a>1 , let P a P^a be the subgroup of P P consisting of all the elements with slopes in a Z a^\mathbb {Z} , and let P Q P^\mathbb {Q} be the subgroup of P P consisting of all the elements with slopes and breaks in Q \mathbb {Q} . We show that the groups P P , P a P^a , P Q P^\mathbb {Q} , as well as Thompson group F F , are invariably generated.

Alsedà–Misiurewicz systems with place-dependent probabilities* *The research was supported by the Polish Ministry of Science and Higher Education ‘Diamond Grant’ 0090/DIA/2017/46.

We consider systems of two specific piecewise linear homeomorphisms of the unit interval, so called Alsedà–Misiurewicz systems, and investigate the basic properties of Markov chains which arise when these two transformations are applied randomly with probabilities depending on the point of the interval. Though this iterated function system is not contracting in average and known methods do not apply, stability and the strong law of large numbers are proven.

Groups of piecewise linear homeomorphisms of flows

To every dynamical system$(X,\varphi )$over a totally disconnected compact space, we associate a left-orderable group$T(\varphi )$. It is defined as a group of homeomorphisms of the suspension of$(X,\varphi )$which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple and, if it is a subshift, then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system$(X,\varphi )$, the group$T(\varphi )$does not have infinite subgroups with Kazhdan's property$(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally, if$(X,\varphi )$has a dense orbit, then the isomorphism type of the group$T(\varphi )$is a complete invariant of flow equivalence of the pair$\{\varphi , \varphi ^{-1}\}$.

On compatible triangulations with a minimum number of Steiner points

Two vertex-labelled polygons are compatible if they have the same clockwise cyclic ordering of vertices. The definition extends to polygonal regions (polygons with holes) and to triangulations—for every face, the clockwise cyclic order of vertices on the boundary must be the same. It is known that every pair of compatible n-vertex polygonal regions can be extended to compatible triangulations by adding O(n2) Steiner points. Furthermore, Ω(n2) Steiner points are sometimes necessary, even for a pair of polygons. Compatible triangulations provide piecewise linear homeomorphisms and are also a crucial first step in morphing planar graph drawings, aka “2D shape animation.” An intriguing open question, first posed by Aronov, Seidel, and Souvaine in 1993, is to decide if two compatible polygons have compatible triangulations with at most k Steiner points. In this paper we prove the problem to be NP-hard for polygons with holes. The question remains open for simple polygons.

Coherent actions by homeomorphisms on the real line or an interval

We study actions of groups by orientation preserving homeomorphisms on R (or an interval) that are minimal, have solvable germs at ±∞ and contain a pair of elements of a certain dynamical type. We call such actions coherent. We establish that such an action is rigid, i.e., any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non-elementary amenable, but satisfies that every proper quotient is solvable. The structure theory we develop allows us to prove a plethora of non-embeddability statements concerning groups of piecewise linear and piecewise projective homeomorphisms. For instance, we demonstrate that any coherent group action G < Horneo+ (R) that produces a nonamenable equivalence relation with respect to the Lebesgue measure satisfies that the underlying group does not embed into Thompson’s group F. This includes all known examples of nonamenable groups that do not contain non abelian free subgroups and act faithfully on the real line by homeomorphisms. We also establish that the Brown-Stein-Thompson groups F(2, pi,…,pn) for n ≥ 1 and p1,…,pn distinct odd primes, do not embed into Thompson’s group F. This answers a question recently raised by C. Bleak, M. Brin and J. Moore.

Uniformly perfect finitely generated simple left orderable groups

We show that the finitely generated simple left orderable groups $G_{\!\unicode[STIX]{x1D70C}}$ constructed by the first two authors in Hyde and Lodha [Finitely generated infinite simple groups of homeomorphisms of the real line. Invent. Math. (2019), doi:10.1007/s00222-019-00880-7] are uniformly perfect—each element in the group can be expressed as a product of three commutators of elements in the group. This implies that the group does not admit any homogeneous quasimorphism. Moreover, any non-trivial action of the group on the circle, which lifts to an action on the real line, admits a global fixed point. It follows that any faithful action on the real line without a global fixed point is globally contracting. This answers Question 4 of the third author [A. Navas. Group actions on 1-manifolds: a list of very concrete open questions. Proceedings of the International Congress of Mathematicians, Vol. 2. Eds. B. Sirakov, P. Ney de Souza and M. Viana. World Scientific, Singapore, 2018, pp, 2029–2056], which asks whether such a group exists. This question has also been answered simultaneously and independently, using completely different methods, by Matte Bon and Triestino [Groups of piecewise linear homeomorphisms of flows. Preprint, 2018, arXiv:1811.12256]. To prove our results, we provide a characterization of elements of the group $G_{\!\unicode[STIX]{x1D70C}}$ which is a useful new tool in the study of these examples.

A finitely presented infinite simple group of homeomorphisms of the circle

We construct a finitely presented, infinite, simple group that acts by homeomorphisms on the circle, but does not admit a non-trivial action by $C^1$-diffeomorphisms on the circle. The group emerges as a group of piecewise projective homeomorphisms of $S^1=\mathbf{R}\cup \{\infty\}$. However, we show that it does not admit a non-trivial action by piecewise linear homeomorphisms of the circle. Another interesting and new feature of this example is that it produces a non amenable orbit equivalence relation with respect to the Lebesgue measure.

Bi-iterative limits of a class of piecewise-linear homeomorphisms

ABSTRACT Let be an increasing homeomorphism with no fixed point such that the derivative exists and 0<d<1. Suppose the Schröder equation has a non-injective regularly varying solution, moreover, for each the function f maps the set linearly onto the set . We investigate necessary and sufficient conditions for the existence of the bi-iterative limits , where .

Reversibility in Groups of Piecewise Linear Homeomorphisms of the Circle

Let PL+(S1) be the group of order preserving piecewise linear homeomorphisms of the circle. An element in PL+(S1) is called reversible in PL+(S1) if it is conjugate to its inverse in PL+(S1). We characterize the reversible elements in PL+(S1). We also perform a sim- ilar characterisation in the full group PL(S1) of piecewise linear home- omorphisms of the circle.

Equivariant smoothing of piecewise linear manifolds

AbstractWe prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly.

A circle diffeomorphism with breaks that is absolutely continuously linearizable

In this paper we answer positively to a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case where its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories and whose invariant measure is absolutely continuous with respect to the Lebesgue measure. The irrational rotation number for our example can be chosen to be a Roth number, but not of bounded type.