Groups Of 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

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].

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.

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$\{\varphi , \varphi ^{-1}\}$.

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.

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.

Explicit left orders on free groups extending the lexicographic order on free monoids

For every finitely generated free group, we construct an explicit left order extending the lexicographic order on the free monoid generated by the positive letters. The order is defined by a left, free action on the orbit of 0 of a free group of piecewise linear homeomorphisms of the line. The membership in the positive cone is decidable in linear time in the length of the input word. The positive cone forms a context-free language closed under word reversal.

Addendum to “Commensurations and subgroups of finite index of Thompson’s groupF”

We show that the abstract commensurator of Thompson's group F is composed of four building blocks: two isomorphism types of simple groups, the multiplicative group of the positive rationals and a cyclic group of order two. The main result establishes the simplicity of a certain group of piecewise linear homeomorphisms of the real line.

Conjugacy in Thompson’s group 𝐹

We complete the program begun by Brin and Squier of characterising conjugacy in Thompson’s group F F using the standard action of F F as a group of piecewise linear homeomorphisms of the unit interval.

Reversible maps and composites of involutions in groups of piecewise linear homeomorphisms of the real line

An element of a group is reversible if it is conjugate to its own inverse, and it is strongly reversible if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be expressed as a composite of two involutions. In this paper the reversible maps, the strongly reversible maps, and those maps that can be expressed as a composite of involutions are determined in certain groups of piecewise linear homeomorphisms of the real line.

Rotation numbers in Thompson–Stein groups and applications

We study the rationality properties of rotation numbers for some groups of piecewise linear homeomorphisms of the circle, the Thompson–Stein groups. We prove that for many Thompson–Stein groups the outer automorphism group has order 2. As another application, we construct Thompson–Stein groups which do not admit non trivial representations in Diff9(S1).

Piecewise Linear Non‐Representability of Out π1M

For every ⩾ 6, examples are given of closed negatively curved riemannian n‐manifolds for which the natural map PLM → Out π1M is not an epimorphism. PLM denotes the group of piecewise linear homeomorphisms of M. 2000 Mathematics Subject Classification 53C20, 57Q25, 57R55.

Inner Amenability of Some Groups of Piecewise-Linear Homeomorphisms of the Real Line

Inner Amenability of Some Groups of Piecewise-Linear Homeomorphisms of the Real Line

The Ubiquity of Thompson's Group F in Groups of Piecewise Linear Homeomorphisms of the Unit Interval

We show that Thompson's group F occurs with great frequency in the group of PL homeomorphisms of the unit interval.

Automorphisms of Generalized Thompson Groups

We look at the automorphisms of Thompson type groups of piecewise linear homeomorphisms of the real line or circle that use slopes that are integral powers of a fixed integer n with n>2. We show that large numbers of "exotic" automorphisms appear---automorphisms that are represented as conjugation by non-PL homeomorphisms of the real line or circle. This is in contrast to the n=2 case where no such automorphisms appear.

Groups of Piecewise-Linear Homeomorphisms with Irrational Slopes

Groups of Piecewise-Linear Homeomorphisms with Irrational Slopes

The chameleon groups of Richard J. Thompson: Automorphisms and dynamics

The automorphism groups of several of Thompson's countable groups of piecewise linear homeomorphisms of the line and circle are computed and it is shown that the outer automorphism groups of these groups are relatively small. These results can be interpreted as stability results for certain structures of PL functions on the circle. Machinery is developed to relate the structures on the circle to corresponding structures on the line.

Groups of Piecewise Linear Homeomorphisms

In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple groups.

Groups of piecewise linear homeomorphisms

In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple groups.

Groups of piecewise linear homeomorphisms of the real line

Groups of piecewise linear homeomorphisms of the real line