Bounded Topological Groups

Each topological group embeds into a duoseparable topological group

A topological group $X$ is called $duoseparable$ if there exists a countable set $S\subseteq X$ such that $SUS=X$ for any neighborhood $U\subseteq X$ of the unit. We construct a functor $F$ assigning to each (abelian) topological group $X$ a duoseparable (abelain-by-cyclic) topological group $FX$, containing an isomorphic copy of $X$. In fact, the functor $F$ is defined on the category of unital topologized magmas. Also we prove that each $σ$-compact locally compact abelian topological group embeds into a duoseparable locally compact abelian-by-countable topological group.

Some properties of Pre-topological groups

In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each $T_{0}$ pre-topological group is regular and every almost topological group is completely regular which extends A.A. Markov's theorem to the class of almost topological groups. Moreover, it is shown that an almost topological group is $τ$-narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less…

Rothberger bounded groups and Ramsey theory

We show that: 1. Rothberger bounded subgroups of sigma-compact groups are characterized by Ramseyan partition relations. 2. For each uncountable cardinal $κ$ there is a ${\sf T}_0$ topological group of cardinality $κ$ such that ONE has a winning strategy in the point-open game on the group and the group is not a subspace of any sigma-compact space. 3. For each uncountable cardinal $κ$ there is a ${\sf T}_0$ topological group of cardinality $κ$ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact.

Countably infinite bounded abelian groups admit no non-discrete locally minimal group topologies

In this note we show that if $G$ is a countably infinite abelian group such that $nG=0$ for some integer $n$, then the only locally minimal group topology on $G$ is the discrete one.

Topological entropy for automorphisms of totally disconnected locally compact groups

We give a "limit-free formula" simplifying the computation of the topological entropy for topological automorphisms of totally disconnected locally compact groups. This result allows us to extend several basic properties of the topological entropy known to hold for compact groups.

On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise

In the paper we study algebraic properties of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ of cofinite partial isometries of the set of positive integers $\mathbb{N}$ with the bounded finite noise $j$. For the monoids $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ we prove counterparts of some classical results of Eberhart and Selden describing the closure of the bicyclic semigroup in a locally compact topological inverse semigroup. In particular we show that for any positive integer $j$ every Hausdorff shift-continuous topology $τ$ on $\mathbf{I}\mathbb{N}_{\infty}^{\bo…

有界型逆半群拓扑全群有限生成子群的增长

Growth of finitely generated subgroups of the topological full groups of inverse semigroups of bounded type

本文证明了在有界型逆半群 $G_0$ 的不可压缩元素集合有限等假设下，其拓扑全群中与 $G_0$ 轨道等价的有限生成子群具有次指数增长，且指数中的幂有界。

Boundedly Simple Groups Have Trivial Bounded Cohomology

We observe that the singular part of the second bounded cohomology group of boundedly simple groups is trivial.

Ramanujan Complexes and bounded degree topological expanders

Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expanders. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov, is whether bounded degree high dimensional expanders, according to these definitions, exist for d >= 2. We …

Relative Bounded Cohomology for Groupoids

We introduce bounded cohomology for (pairs of) groupoids and develop homological algebra to deal with it. We generalise results of Ivanov, Frigerio and Pagliantini to this setting and show that (under topological conditions) the bounded cohomology of a pair of topological spaces is isometrically isomorphic to the bounded cohomology of the pair of fundamental groupoids. Furthermore, we prove that bounded cohomology relative to an amenable groupoid is isometrically isomorphic to the bounded cohomology of the ambient groupoid.

Full groups of bounded automaton groups

We show that every bounded automaton group can be embedded in a finitely generated, simple amenable group. The proof is based on the study of the topological full groups associated to the Schreier dynamical system of the mother groups. We also show that if $\mathcal{G}$ is a minimal étale groupoid with unit space the Cantor set, the group $[[\mathcal{G}]]_t$ generated by all torsion elements in the topological full group has simple commutator subgroup.

A note on topological amenability

We point out a simple characterisation of topological amenability in terms of bounded cohomology, following Johnson's reformulation of amenability.

A note on bounded-cohomological dimension of discrete groups

Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to groups with infinite bounded-cohomological dimension, and we will provide new examples of groups with bounded-cohomological dimension equal to 0. In particular, we will prove that every group functorially embeds into an acyclic group with trivial bounded cohomology.

局部紧群在拓扑向量空间上的表示之分解与有界性

Factorization and boundedness for representations of locally compact groups on topological vector spaces

本文研究局部紧群在拓扑向量空间上的表示，证明了态射分解、极大殆周期性刻画、von Neumann核恢复及紧群表示有限分解，推广了多项已知结果。

Topological groups all continuous automorphisms of which are open

A topological space is reversible if each continuous bijection of it onto itself is open. We introduce an analogue of this notion in the category of topological groups: A topological group G is g-reversible if every continuous automorphism of G (=continuous isomorphism of G onto itself) is open. The class of g-reversible groups contains Polish groups, locally compact sigma-compact groups, minimal groups, abelian groups with the Bohr topology, and reversible topological groups. We prove that subgroups of R^n are g-reversible, for every positive integer n. An example of a compact (so reversible)…

Cohomology of profinite groups of bounded rank

We generalise to profinite groups some of our previous results on the cohomology of pro-p groups of bounded sectional p-rank.

The second bounded cohomology of a group with infinitely many ends

We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we have a group with infinitely many ends has infinite dimensional second bounded cohomology.

Bounded Cohomology of Groups acting on Cantor sets

We study the bounded cohomology of certain groups acting on the Cantor set. More specifically, we consider the full group of homeomorphisms of the Cantor set as well as Thompson's group $V$. We prove that both of these groups are boundedly acyclic, that is the bounded cohomology with trivial real coefficients vanishes in positive degrees. Combining this result with the already established $\mathbb{Z}$-acyclicity of Thompson's group $V$, will make $V$ the first example of a finitely generated group, in fact the first example of a group of type $F_\infty$, which is universally boundedly acyclic.…

Topological Full Groups

Topological full groups originated from the theory of topological dynamical systems and have been having considerable impact on group theory in recent years. This text represents an introduction/survey on topological full groups. After development of the theoretical and historical background, it gives an account of their significance in topological dynamics and discusses their group theoretical aspects.

Finite index subgroups in Chevalley groups are bounded: an addendum to "On bi-invariant word metrics"

We prove that finite index subgroups in S-arithmetic Chevalley groups are bounded.

Strongly bounded groups of various cardinalities

Strongly bounded groups are those groups for which every action by isometries on a metric space has orbits of finite diameter. Many groups have been shown to have this property, and all the known infinite examples so far have cardinality at least $2^{\aleph_0}$. We produce examples of strongly bounded groups of many cardinalities, including $\aleph_1$, answering a question of Yves de Cornulier [4]. In fact, any infinite group embeds as a subgroup of a strongly bounded group which is, at most, two cardinalities larger.

On the bounded cohomology of semi-simple groups, S-arithmetic groups and products

We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices.

On elementary amenable bounded automata groups

There are several natural families of groups acting on rooted trees for which every member is known to be amenable. It is, however, unclear what the elementary amenable members of these families look like. Towards clarifying this situation, we here study elementary amenable bounded automata groups. We are able to isolate the elementary amenable bounded automata groups in three natural subclasses of bounded automata groups. In particular, we show that iterated monodromy groups of post-critically finite polynomials are either virtually abelian or not elementary amenable.

The Ext group in the categories of topological abelian groups and topological vector spaces

This paper is devoted to the study of the group ${\rm Ext} (G,H)$ of all extensions of topological abelian groups $0\to H\to X\to G\to 0$ and the group ${\rm Ext}_{TVS} (Z,Y)$ of all extensions of topological vector spaces $0\to Y\to X\to Z\to 0$. We focus on their behaviour under taking products, countable coproducts, dense subgroups and open subgroups. Finally, we apply the obtained properties to formulate in a more general setting some known results in the category of locally compact abelian groups and to determine conditions in which being a topological vector space is a three space proper…

On the topological full group containing the Grigorchuk group

We consider the topological full group of a substitution subshift induced by a substitution $a\to aca$, $b\to d$, $c\to b$, $d\to c$. This group is interesting since the Grigorchuk group naturally embeds into it. We show that the topological full group is finitely generated and give a simple generating set for it.

标记群空间中的有界简洁性

Bounded conciseness in the space of marked groups

本文证明了有界简洁性在标记群空间中构成闭性质，并据此重新表述了Fernández-Alcober与Shumyatsky关于剩余有限群类中简洁性的猜想。

On topological rough groups

In this paper, we give an introduction for rough groups and rough homomorphisms. Then we present some properties related to topological rough subgroups and rough subsets. We construct the product of topological rough groups and give an illustrated example. Then, we define topological rough group homomorphisms and topological rough group homeomorphisms. Finally, we introduce a rough action, a rough homogenous space and a rough kernel.

Embedding topological semigroups into the hyperspaces over topological groups

We study algebraic and topological properties of subsemigroups of the hyperspace exp(G) of non-empty compact subsets of a topological group G endowed with the Vietoris topology and the natural semigroup operation. On this base we prove that a compact Clifford topological semigroup S is topologically isomorphic to a subsemigroup of exp(G) for a suitable topological group G if and only if S is a topological inverse semigroup with zero-dimensional idempotent semilattice.

On representations of direct products and the bounded generation property of branch groups

We prove that the minimal representation dimension of a direct product $G$ of non-abelian groups $G_1,\ldots,G_n$ is bounded below by $n+1$ and thereby answer a question of Abért. If each $G_i$ is moreover non-solvable, then this lower bound can be improved to be $2n$. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups we show that branch groups cannot be boundedly generated.

Large scale properties for bounded automata groups

In this paper, we study some large scale properties of the mother groups of bounded automata groups. First we give two methods to prove every mother group has infinite asymptotic dimension. Then we study the decomposition complexity of certain subgroup in the mother group. We prove the subgroup belongs to $\mathcal{D}_ω$.

On subgroups of saturated or totally bounded paratopological groups

A paratopological group $G$ is saturated if the inverse $U^{-1}$ of each non-empty set $U\subset G$ has non-empty interior. It is shown that a [first-countable] paratopological group $H$ is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if $H$ admits a continuous bijective homomorphism onto a (totally bounded) [abelian] topological group $G$ [such that for each neighborhood $U\subset H$ of the unit $e$ there is a closed subset $F\subset G$ with $e\in h^{-1}(F)\subset U$]. As an application we construct a paratopological group whose character exce…

Pseudocompact paratopological groups that are topological

We obtain necessary and sufficient conditions when a pseudocompact paratopological group is topological. (2-)pseudocompact and countably compact paratopological groups that are not topological are constructed. It is proved that each 2-pseudocompact paratopological group is pseudocompact and that each Hausdorff σ-compact pseudocompact paratopological group is a compact topological group. Our particular attention is devoted to periodic and topologically periodic pseudocompact paratopological groups.

Bounded rank subgroups of Coxeter groups, Artin groups and one-relator groups with torsion

We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.

Bounded cohomology and binate groups

A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first non-amenable examples were the group of compactly supported homeomorphisms of $\mathbb{R}^n$ (Matsumoto--Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic. We obtain several new examples of boundedly acyclic groups as well as comp…

Strongly bounded groups and infinite powers of finite groups

We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the so-called uncountable strong cofinality, recently introduced by G. Bergman. Our main result is that G^I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that omega_1-existentially closed groups are strongly bounded.

Bounded Automata Groups are co-ET0L

Holt and Röver proved that finitely generated bounded automata groups have indexed co-word problem. Here we sharpen this result to show they are in fact co-ET0L.

Low-dimensional bounded cohomology and extensions of groups

Bounded cohomology of groups was first studied by Gromov in 1982. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic `non-positively curved' groups. On the other hand, there is a well-known interpretation of ordinary group cohomology in dimension 2 and 3 in terms of group extensions. The aim of this paper is to make this interpretation available for bounded group cohomology. This will involve quasihomomorphisms as defined and studied by Fujiwara and Kapovich.

Bounded cohomology, cross ratios and cocycles

We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a proper hyperbolic geodesic metric space.

强1有界的内可 amenable 群

Strongly 1-bounded inner amenable groups

本文研究了有限生成icc内可 amenable群，证明其生成强1有界的II_1因子。

Bounded cohomology of finitely presented groups: vanishing, non-vanishing, and computability

We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-amenable boundedly acyclic groups and construct a finitely presented non-amenable boundedly acyclic group. On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has uncountable dimension in all degrees greater than or equal to~$2$, and a concrete finitely presented one. Countable non-amenable groups with these two extre…

Strong and uniform boundedness of groups

A group G is called bounded if every conjugation-invariant norm on G has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.

Amenability of Bounded Automata Groups on Infinite Alphabets

We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire functions.

拓扑Kac-Moody群的高阶同伦群、Whitehead塔与弦群

Higher homotopy groups of topological Kac-Moody groups, Whitehead tower, and string groups

本文证明拓扑分裂实Kac-Moody群的同伦群可数，从而推出其Whitehead塔存在，并给出$E_n$系列弦群的构造方法。

Bounded cohomology of discrete groups

Bounded cohomology of groups was first defined by Johnson and Trauber during the seventies in the context of Banach algebras. As an independent and very active research field, however, bounded cohomology started to develop in 1982, thanks to the pioneering paper "Volume and Bounded Cohomology" by M. Gromov, where the definition of bounded cohomology was extended to deal also with topological spaces. The aim of this monograph is to provide an introduction to bounded cohomology of discrete groups and of topological spaces. We also describe some applications of the theory to related active resear…

The isometry group of the bounded Urysohn space is simple

We show that the isometry group of the bounded Urysohn space is simple. This extends previous work by the authors.

Bounded generation by root elements for Chevalley groups defined over rings of integers of function fields with an application in strong boundedness

Bounded generation by root elements is a property which has been widely studied for various types of linear algebraic groups defined over rings of integers in algebraic number fields. However, when considering global function fields, there are not many results beyond the treatment of special cases due to Nica and Queen. In this paper, we use model theoretic methods due to Carter, Keller and Paige written up by Morris to prove bounded generation by root elements for simply connected, split Chevalley groups defined over the ring of all integers in a global function field. We further apply this b…

Space of group orderings, quasi morphisms and bounded cohomology

For a group $G$, we construct a quasi morphism from its left orderings and the map from the space of left orderings to the second bounded cohomology. We show that these maps reflect various properties of the group orderings.

可实现为有界域自同构群的非线性李群

Non-linear Lie groups that can be realized as automorphism groups of bounded domains

本文研究李群能否成为 C^n 有界域双全纯自同构群，在1990年连通线性李群结论基础上，给出满足该实现性质的非线性李群例子。

On topological complexity of hyperbolic groups

We show that the topological complexity of a finitely generated torsion free hyperbolic group $π$ with $\cdπ=n$ equals $2n$.

Uniformly bounded representations and exact groups

We characterize groups with Guoliang Yu's property A (i.e., exact groups) by the existence of a family of uniformly bounded representations which approximate the trivial representation.

无限秩自由群的自同构群是粗有界的

The Automorphism Group of the Infinite-Rank Free Group is Coarsely Bounded

本文证明了可数无限秩自由群的完全自同构群和外部自同构群是粗有界的，即它们在度量空间上不存在具有无界轨道的连续作用，且具有一个点的拟等距类型。

Lamplighters and the bounded cohomology of Thompson's group

We prove the vanishing of the bounded cohomology of lamplighter groups for a wide range of coefficients. This implies the same vanishing for a number of groups with self-similarity properties, such as Thompson's group F. In particular, these groups are boundedly acyclic. Our method is ergodic and applies to large transformation groups where the Mather-Matsumoto-Morita method sometimes fails because not all are acyclic in the usual sense.

On the bounded cohomology of Lie groups

We show that each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle if and only if the radical of G is linear. This leads to a generalization of Gromov's boundedness theorem on characteristic classes of flat bundles.

有界哈德维格数图的图自同构群

Automorphism groups of graphs of bounded Hadwiger number

本文研究有界哈德维格数有限图的自同构群结构，通过分析边传递图证明非阿贝尔合成因子阶有界，并表明这类自同构群可由阿贝尔群、对称群和有界阶群反复扩张得到。