Weak Identities in finitely generated groups

右理想有限生成的半群

Semigroups whose right ideals are finitely generated

本文研究右理想有限生成的半群（弱右诺特半群），给出其等价刻画，并探讨其在商、子半群、直积、正则半群及强半格等构造下的性质，最终证明交换半群弱右诺特的充要条件。

On the Bieri-Neumann-Strebel-Renz invariants of the weak commutativity construction $\X(G)$

For a finitely generated group $G$ we calculate the Bieri-Neumann-Strebel-Renz invariant $Σ^1(\X(G))$ for the weak commutativity construction $\X(G)$. Identifying $S(\X(G))$ with $S(\X(G) / W(G))$ we show $Σ^2(\X(G),\Z) \subseteq Σ^2(\X(G)/ W(G),\Z)$ and $Σ^2(\X(G)) \subseteq $ $ Σ^2(\X(G)/ W(G))$ that are equalities when $W(G)$ is finitely generated and we explicitly calculate $Σ^2(\X(G)/ W(G),\Z)$ and $ Σ^2(\X(G)/ W(G))$ in terms of the $Σ$-invariants of $G$. We calculate completely the $Σ$-invariants in dimensions 1 and 2 of the group $ν(G)$ and show that if $G$ is finitely generated group …

Weak commutativity and finiteness properties of groups

We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from this construction: $\mathfrak{X}(G)$ is finitely presented if and only if $G$ is finitely presented, but if $F$ is a non-abelian free group of finite rank then $\mathfrak{X}(F)$ has a subgroup of finite index whose third homology is not finitely generated.

Quantifying lawlessness in finitely generated groups

We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" $\mathcal{A}_Γ : \mathbb{N} \rightarrow \mathbb{N}$. We show that $\mathcal{A}_Γ$ is bounded iff $Γ$ has a nonabelian free subgroup. By contrast we construct, for any nondecreasing unbounded function $f: \mathbb{N} \rightarrow \mathbb{N}$, an elementary amenable lawless groups for which $\mathcal{A}_Γ$ grows more slowly that $f$. We produce torsion lawless groups for which $\mathcal{A}_Γ$ is at least linear using Golod-Shafarevich theory, and give some upper bounds on $…

Embedding free Burnside groups in finitely presented groups

We construct an embedding of a free Burnside group $B(m,n)$ of odd $n > 2^{48}$ and rank $m >1$ in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented non-amenable groups without noncyclic free subgroups (which provides a finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent $n \gg 1$ which are not locally finite.

The isoperimetric spectrum of finitely presented groups

The isoperimeric spectrum consists of all real positive numbers $α$ such that $O(n^α)$ is the Dehn function of a finitely presented group. In this note we show how a recent result of Olshanskii completes the description of the isoperimetric spectrum modulo the celebrated Computer Science conjecture (and one of the seven Millennium Problems) $\mathbf{P=NP}$ and even a formally weaker conjecture.

具有强不动点性质和性质（T）的有限呈示群的例子

Examples of finitely presented groups with strong fixed point properties and property (T)

本文通过推广Hall嵌入定理，构造了一个具有性质（T）但无法在合理空间上作用的有限呈示群，展示了性质（T）与简单性要求之间的权衡。

Every finitely generated group is weakly exact

We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf $G$-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf $G$-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on $\ell_{\infty}$-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.

无限群中的广义 torsion 元素

Generalized torsion elements in infinite groups

研究无限群中广义 torsion 元素的性质，证明了有限生成交换陪有限群中广义 torsion 元素的判定条件，并给出了相关群的定量结果和例子。

On Systems of Equations over Free Products of Groups

Using an analogue of Makanin-Razborov diagrams, we give a description of the solution set of systems of equations over an equationally Noetherian free product of groups $G$. Equivalently, we give a parametrisation of the set $Hom(H, G)$ of all homomorphisms from a finitely generated group $H$ to $G$. Furthermore, we show that every algebraic set over $G$ can be decomposed as a union of finitely many images of algebraic sets of NTQ systems. If the universal Horn theory of $G$ (the theory of quasi-identities) is decidable, then our constructions are effective.

Coherency and constructions for monoids

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck--Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of $S$ to be finitely generated.

弱双曲群中的边界动力学、三重传递性和混合恒等式

Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups

研究在δ-双曲空间上具有一般类型作用且诱导作用忠实于Gromov边界极限集的群，根据拓扑自由性划分亚类，证明拓扑自由等价于无混合恒等式，并应用于Kac-Moody群。

The subgroup identification problem for finitely presented groups

We introduce the subgroup identification problem, and show that there is a finitely presented group G for which it is unsolvable, and that it is uniformly solvable in the class of finitely presented locally Hopfian groups. This is done as an investigation into the difference between strong and weak effective coherence for finitely presented groups.

有限生成双可序群的同构问题

The isomorphism problem for finitely generated bi-orderable groups

从描述集合论角度研究有限生成可序群的分类问题，证明有限生成双可序群的同构关系是弱普遍的。

有限生成可解群中的均匀几乎平坦性

Uniformly Almost Flatness and Solubility in Finitely Generated Groups

研究有限生成可解群虚拟 nil 潜在性与有限陪集空间直径的多项式下界关系，扩展了与 Tointon 合作的先前工作。

骰子周期群

Dice periodic groups

本文构造了一族有限生成的无限周期群，以四面体群为基本例子，并推广提出骰子群，给出了其成为周期群的弱条件及元素阶的素数乘积性质。

变换与划分幺半群的相干性质

Coherency properties for monoids of transformations and partitions

本文证明无限集上的全/部分变换幺半群、对称逆幺半群和划分幺半群均为弱右相干但非右相干，并通过对偶定义给出左相干对应结果，借助一个不可嵌入的逆半群表示证明非相干性。

On the wgsc and qsf tameness conditions for finitely presented groups

A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc i.e. it has an exhaustion by compact connected and simply connected sub-polyhedra. We show that this condition is almost-equivalent to Brick's qsf property, which amounts to finding an exhaustion approximable by finite simply connected complexes, and also to the tame combability introduced and studied by Mihalik and Tschantz. We further observe that a number of standard constructions in group theory yield qsf groups and analyze sp…

紧阿贝尔群自同构的有限支撑不变测度的稠密性

Density of finitely supported invariant measures for automorphisms of compact abelian groups

本文证明满足降链条件的紧可度量阿贝尔群自同构的有限支撑不变测度在弱-*拓扑下稠密，并利用specification性质变体得到Hilbert-Schmidt稳定性与Livshitz型定理等推论。