On approximation of topological groups by finite algebraic systems. II

代数上同调群、代数基本群和稳定自由模

Algebraic cohomotopy groups, algebraic fundamental groups and stably free modules

本文研究了代数上同调群与稳定自由模的关系，证明了拓扑结果的代数对应，并展示了非自由稳定自由模的例子。

Some algebraic and topological properties of subspaces of (pre)orders

We study algebraic and topological properties of subsets of preorders on a group. In particular we study properties of the composition of two preorders, generalize a topological theorem of \cite{S} in the case of standard orders and show the same theorem in the case of standard preorders. We also show a property of standard valuations.

Topological generation of exceptional algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field $k$ and let $C_1, \ldots, C_t$ be non-central conjugacy classes in $G$. In this paper, we consider the problem of determining whether there exist $g_i \in C_i$ such that $\langle g_1, \ldots, g_t \rangle$ is Zariski dense in $G$. First we establish a general result, which shows that if $Ω$ is an irreducible subvariety of $G^t$, then the set of tuples in $Ω$ generating a dense subgroup of $G$ is either empty or dense in $Ω$. In the special case $Ω= C_1 \times \cdots \times C_t$, by considering the dimensions of fixed point s…

Algebraic, combinatorial and topological properties of singular virtual braid monoids

In this paper we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side we state the relations between classical and virtual singular objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove that there is a bijection between singular abstract braids, horizontal Gauss diagrams and singular virtual braids, in particular using horizontal Gauss diagrams we obtain a presentation of the singular pure virtual braid monoid.

奇异双群的代数与拓扑方面及其表示

Algebraic and topological aspects of the singular twin group and its representations

本文从代数和拓扑角度研究了奇异双群及其表示，分类了复齐性2-局部表示并探讨了其不可约性条件。

Algebraic entropy on topologically quasihamiltonian groups

We study the algebraic entropy of continuous endomorphisms of compactly covered, locally compact, topologically quasihamiltonian groups. We provide a Limit-free formula which helps us to simplify the computations of this entropy. Moreover, several Addition Theorems are given. In particular, we prove that the Addition Theorem holds for endomorphisms of quasihamiltonian torsion FC-groups (e.g., Hamiltonian groups).

Some Properties of the Schur Multiplier with Algebraic Topological Approach

In this paper, using a relationship between the Schur multiplier of a group $G$, the fundamental group, and the second homology group of the Eilenberg-MacLane space of $G$, we present new proofs for some famous properties of the Schur multiplier and its structure for the free product and the direct product with an algebraic topological approach. Also, we try to find the Schur multiplier of a free amalgamated product in terms of Schur multipliers of its factors, which is a new result.

Surjunctivity and topological rigidity of algebraic dynamical systems

Let $X$ be a compact metrizable group and $Γ$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,Γ)$ is surjunctive, i.e., every injective continuous map $τ\colon X \to X$ commuting with the action of $Γ$ is surjective.

Computability Models: Algebraic, Topological and Geometric Algorithms

The discreteness problem for finitely generated subgroups of $PSL(2,\mathbb{R})$ and $PSL(2,\mathbb{C})$ is a long-standing open problem. In this paper we consider whether or not this problem is decidable by an algorithm. Our main result is that the answer depends upon what model of computation is chosen. Since our discussion involves the disparate topics of computability theory and group theory, we include substantial background material.

阿贝尔群上的代数扩张性

Algebraic expansivity on abelian groups

本文研究了阿贝尔群上端态射的代数扩张性，探讨了其基本性质、与Weiss代数熵的关系，并证明正向扩张满射仅限于有限系统，同时通过庞特里亚金对偶性展示了其与拓扑动力学的联系。

Topological and algebraic pullback functors

We give algebraic equivalents for certain desirable properties of pullback functors on categories of coverings and group sets, namely nullity zero, essential injectivity, and essential surjectivity. Nullity zero turns out to be equivalent to the notion of a contranormal subgroup. We observe a Tannakian-like phenomenon with essential injectivity. Essential surjectivity is intimately related to Zappa-Sz{é}p products. We include several examples, and some open questions.

On strong approximation for algebraic groups

This is an expanded version of my talk given at the workshop "Hot Topics: Thin Groups and Super-strong Approximation" (MSRI, Berkeley, February 6-10, 2012).

Adjoint entropy vs Topological entropy

Recently the adjoint algebraic entropy of endomorphisms of abelian groups was introduced and studied. We generalize the notion of adjoint entropy to continuous endomorphisms of topological abelian groups. Indeed, the adjoint algebraic entropy is defined using the family of all finite-index subgroups, while we take only the subfamily of all open finite-index subgroups to define the topological adjoint entropy. This allows us to compare the (topological) adjoint entropy with the known topological entropy of continuous endomorphisms of compact abelian groups. In particular, the topological adjoin…

On approximation of topological groups by finite algebraic systems

It is known that locally compact groups approximable by finite ones are unimodular, but this condition is not sufficient, for example, the simple Lie groups are not approximable by finite ones as topological groups. In this paper the approximations of locally compact groups by more general finite algebraic systems are investigated. It is proved that the approximation of locally compact groups by finite semigroups is equivalent to approximation by finite groups and thus not all locally compact groups are approximable by finite semigroups. We prove that any locally compact group is approximable …

Topological Entropy and Algebraic Entropy for group endomorphisms

The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Math…

丰富群胚、拓扑全群、代数K-理论谱与无穷环空间

Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

本文受Higman-Thompson群同调研究启发，通过构造紧开双截面的小置换范畴，建立了丰富群胚、拓扑全群、代数K-理论谱与无穷环空间间的联系，并应用于计算同调不变量、证明消没及无环性结果，最终证明了Matui的AH猜想。

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

We continue our study of when topological and measure-theoretic entropy agree for algebraic action of sofic groups. Specifically, we provide a new abstract method to prove that an algebraic action is strongly sofic. The method is based on passing to a "limiting object" for sequences of measures which are asymptotically supported on topological microstates. This limiting object has a natural poset structure that we are able to exploit to prove a max-min principle: if the sofic approximation has ergodic centralizer, then the largest subgroup on which the action is a local weak$^{*}$-limit of mea…

Algebraic Yuzvinski Formula

In 1965 Adler, Konheim and McAndrew defined the topological entropy for continuous self-maps of compact spaces. Topological entropy is very well-understood for endomorphisms of compact Abelian groups. A fundamental result in this context is the so-called Yuzvinski Formula, showing that the value of the topological entropy of a full solenoidal automorphism coincides with the Mahler measure of its characteristic polynomial. In two papers of 1979 and 1981 Peters gave a different definition of entropy for automorphisms of locally compact Abelian groups. This notion has been appropriately modified …