Group Of Homeomorphisms Research Articles

The homeomorphism group of the first uncountable ordinal

We show that the topology of pointwise convergence on scattered spaces is compatible with the group structure of their homeomorphism group. We then establish a few topological properties of the homeomorphism group of the first uncountable ordinal, such as amenability and Roelcke-precompactness.

Dynamics and structure of groups of homeomorphisms of scattered spaces

We study the topological structure and the topological dynamics of groups of homeomorphisms of scattered spaces. For a large class of them (including the homeomorphism group of any ordinal space or of any locally compact scattered space), we establish Roelcke-precompactness and amenability, classify all closed normal subgroups and compute the universal minimal flow. As a by-product, we classify up to isomorphism the homeomorphism groups of compact ordinal spaces.

Automorphism towers of groups of homeomorphisms of Cantor space

We show that for any full and sufficiently transitive (i.e., flexible) group G of homeomorphisms of Cantor space, Aut(Aut(G)) = Aut(G). This class contains many generalisations of the Higman-Thompson g roups Gn,r, and the Rational group $${{\cal R}_2}$$ ℛ 2 of Grigorchuk, Nekrashevych, and Sushchanskiĭ. We also demonstrate that for generalisations Tn,r of R. Thompson’s group T, Aut(Aut(Tn,r)) = Aut(Tn,r). In the case of the groups Gn,r and Tn,r our results extend results of Brin and Guzmán for Thompson’s group T, and generalisations of Thompson’s group F.

On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees

Consider an infinite homogeneous tree \mathcal{T}_n of valence n+1 , its group Aut (\mathcal{T}_n) of automorphisms, and the group Hier (\mathcal{T}_n) of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of \mathcal{T}_n that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut (\mathcal{T}_n) is spherical in Hier (\mathcal{T}_n) , i.e., any irreducible unitary representation of Hier (\mathcal{T}_n) contains at most one Aut (\mathcal{T}_n) -fixed vector. We present a combinatorial description of the space of double cosets of Hier (\mathcal{T}_n) with respect to Aut (\mathcal{T}_n) and construct a “new” family of spherical representations of Hier (\mathcal{T}_n) . We also show that the Thompson group Th has PSL (2,\mathbb{Z}) -spherical unitary representations.

The Kirch space is topologically rigid

The Golomb space (resp. the Kirch space) is the set N of positive integers endowed with the topology generated by the base consisting of arithmetic progressions a+bN0={a+bn:n≥0} where a,b∈N and b is a (square-free) number, coprime with a. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected). By a recent result of Banakh, Spirito and Turek, the Golomb space has trivial homeomorphism group and hence is topologically rigid. In this paper we prove the topological rigidity of the Kirch space.

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) .

Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.

On C*-algebras of singular integral operators with PQC coefficients and shifts with fixed points

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the -algebra generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator and by the unitary weighted shift operators , acting on the space over the unit circle . Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms with finite sets of discontinuities for their derivatives , which acts topologically freely on , where is the interior of the nonempty closed set composed by all common fixed points for all shifts , with boundary of zero Lebesgue measure. A Fredholm symbol calculus for the -algebra is constructed and a Fredholm criterion for the operators is established.

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.

On James Hyde’s example of non-orderable subgroup of $\mathrm{Homeo}(D,\partial D)$

In Ann. Math. 190 (2019), 657–661, James Hyde presented the first example of non-left-orderable, finitely generated subgroup of \mathrm{Homeo}(D,\partial D) , the group of homeomorphisms of the disk fixing the boundary. This implies that the group \mathrm{Homeo}(D,\partial D) itself is not left-orderable. We revisit the construction, and present a slightly different proof of purely dynamical flavor, avoiding direct references to properties of left-orders. Our approach allows to solve the analogue problem for actions on the circle.

Discrete locally finite full groups of Cantor set homeomorphisms

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological full groups). We determine that any piecewise full group that is locally compact in the compact-open topology on the group of self-homeomorphisms of the Cantor set must be uniformly discrete, in a precise sense that we introduce here. Uniformly discrete groups of self-homeomorphisms of the Cantor set are in particular countable, locally finite, residually finite and discrete in the compact-open topology. The resulting piecewise full groups form a subclass of the ample groups introduced by Krieger. We determine the structure of these groups by means of their Bratteli diagrams and associated dimension ranges ($K_0$ groups). We show through an example that not all uniformly discrete piecewise full groups are subgroups of the ``obvious'' ones, namely, piecewise full groups of finite groups.

On uniqueness of invariant measures for random walks on

AbstractWe consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$ . In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was studied by Choquet and Deny [Sur l’équation de convolution $\mu = \mu * \sigma $ . C. R. Acad. Sci. Paris250 (1960), 799–801] in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on a broader class of systems satisfying the conditions of recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$ . Our work can be viewed as following on from Babillot et al [The random difference equation $X_n=A_n X_{n-1}+B_n$ in the critical case. Ann. Probab.25(1) (1997), 478–493] and Deroin et al [Symmetric random walk on $\mathrm {HOMEO}^{+}(\mathbb {R})$ . Ann. Probab.41(3B) (2013), 2066–2089].

Fuzzy Convergence, Fuzzy Neighborhood Convergence and I-Tolerance Structures for Groups

We introduce a category of fuzzy convergence groups, FCONVGRP a subcategory of the category of fuzzy convergence spaces, FCONV. Viewing [Formula: see text] as a complete Heyting algebra, we prove that the category of [Formula: see text]-tolerance groups, [Formula: see text]-TOLGRP is isomorphic to a subcategory of FCONVGRP. Since FCONV is a topological universe, and thereby possesses function space structure, upon invoking this, we are able, among others, to show that FCONVGRP is topological, and more importantly, it enables us to obtain a compatible fuzzy convergence function space structure on group of homeomorphisms. It is noticeable, however, that the category of fuzzy neighborhood convergence groups, FNCONVGRP — a supercategory of the well-known category FNS, of fuzzy neighborhood spaces, as well as the category of fuzzy neighborhood groups, FNGRP — a subcategory of FNCONVGRP exhibit nice relationships with FCONVGRP. It is important to note that the objects of FCONVGRP are homogeneous, this paves the way to present two pertinent characterization theorems on fuzzy convergence groups. Finally, introducing a category PSTOPGRP, of pseudotopological groups, we reveal the embeddings of FTOPGRP and PSTOPGRP into FCONVGRP.

The Structure of Homeomorphism and Diffeomorphism Groups

The Structure of Homeomorphism and Diffeomorphism Groups

Isometry groups of infinite-genus hyperbolic surfaces

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannian metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

Constraints on families of smooth 4–manifolds from Bauer–Furuta invariants

We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's $10/8$ theorem. As an application we construct examples of continuous $\mathbb{Z}_p$-actions for any odd prime $p$, which can not be realised smoothly. As a second application we show that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a weak homotopy equivalence for any compact, smooth, simply-connected indefinite $4$-manifold with signature of absolute value greater than $8$.

Universal minimal flows of homeomorphism groups of high-dimensional manifolds are not metrizable

Answering a question of Uspenskij, we prove that if X is a closed manifold of dimension 2 or higher or the Hilbert cube, then the universal minimal flow of $${\text {Homeo}}(X)$$ is not metrizable. In dimension 3 or higher, we also show that the minimal $${\text {Homeo}}(X)$$ -flow consisting of all maximal, connected chains in X has meager orbits.

Minimal flows with arbitrary centralizer

AbstractGiven a G-flow X, let $\mathrm{Aut}(G, X)$ , or simply $\mathrm{Aut}(X)$ , denote the group of homeomorphisms of X which commute with the G action. We show that for any pair of countable groups G and H with G infinite, there is a minimal, free, Cantor G-flow X so that H embeds into $\mathrm{Aut}(X)$ . This generalizes results of [2, 7].

Closed orbits and the hyperspace of [formula omitted]-homogeneous continua

Given a metric continuum X, we say that X is 12-homogeneous provided that X has exactly two orbits under the action of the group of homeomorphisms of X onto itself. In this paper, we introduce the hyperspace C12(X) consisting of all 12-homogeneous subcontinua of X. We study compactness and connectedness of C12(X) for some classes of continua, mainly on finite graphs and dendroids, and we characterize the arc and the simple closed curve by using the structure of this hyperspace. Also, we show that every decomposable irreducible 12-homogeneous continuum has one closed orbit.

Topological realizations of groups in Alexandroff spaces

We prove that every group can be realized as the homeomorphism group and as the group of (pointed) homotopy classes of (pointed) self-homotopy equivalences of infinitely many non-homotopy-equivalent Alexandroff spaces.