自由群中的平移等价元

Groups elementarily equivalent to a free nilpotent group of finite rank

In this paper we give a complete algebraic description of groups elementarily equivalent to a given free nilpotent group of finite rank.

A translation of G. S. Makanin's 1966 Ph.D. thesis "On the Identity Problem for Finitely Presented Groups and Semigroups"

This is an English translation of the thesis written by G. S. Makanin for the degree of Candidate of Physical and Mathematical Sciences (equivalent to a Ph.D.), originally submitted to the Steklov Mathematical Institute in 1966. The original language is Russian. The named supervisors are A. A. Markov and S. I. Adian.

群Elementarily Equivalent到有限秩的自由2-幂零群幂零群

Groups Elementarily Equivalent to a Free 2-nilpotent Group of Finite Rank

本文给出了初等等价于自由幂零群$G$的任意有限秩群的一个刻划.

初等等价于有限生成自由亚幂零群的群

Groups elementary equivalent to finitely generated free metabelian

本文描述了初等等价于具有n个生成元的自由亚幂零群的群，并探讨了亚幂零群中自然出现的指数运算。

A counterexample to the easy direction of the geometric Gersten conjecture

For finitely generated groups H and G, equipped with word metrics, a translation-like action of H on G is a free action such that each element of H acts by a map which has finite distance from the identity map in the uniform metric. For example, if H is a subgroup of G, then right translation by elements of H yields a translation-like action of H on G. Whyte asked whether a group with no translation-like action by a Baumslag-Solitar group must be hyperbolic, where the free abelian group of rank 2 is understood to be a Baumslag-Solitar group. We show that the converse of this conjecture is fals…

Upper-modular and related elements of the lattice of commutative semigroup varieties

We completely determine upper-modular, codistributive and costandard elements in the lattice of all commutative semigroup varieties. In particular, we prove that the properties of being upper-modular and codistributive elements in the mentioned lattice are equivalent. Moreover, in the nil-case the properties of being elements of all three types turn out to be equivalent.

两生成群作用于树上的翻译长度公式

Translation length formula for two-generated groups acting on trees

本文提出了两生成群作用于树上的翻译长度公式，并通过伪长度公理推导出相关结论，为自由群的自同构轨道研究提供新方法。

On groups elementarily equivalent to a group of triangular matrices $T_n(R)$

In this paper we investigate the structure of groups elementarily equivalent to the group $T_n(R)$ of all invertible upper triangular $n\times n$ matrices, where $n\geq 3$ and $R$ is a characteristic zero integral domain. In particular we give both necessary and sufficient conditions for a group being elementarily equivalent to $T_n(R)$ where $R$ is a characteristic zero algebraically closed field, a real closed field, a number field, or the ring of integers of a number field.

与经典矩阵群初等等价的群

Groups elementarily equivalent to the classical matrix groups

本文刻画了在域上且维数至少为3时，与经典矩阵群一阶等价的所有群的结构。

元群与巴乌斯拉-索拉特群的正则双可解释性

Groups elementarily equivalent to metabelian Baumslag$-$Solitar groups and regular bi-interpretability

本文证明了巴乌斯拉-索拉特群与整数环的正则双可解释性，并描述了与该群元素等价的所有群。

周期阿贝尔群的自同态环与自同构群的初等等价

Elementary equivalence of endomorphism rings and automorphism groups of periodic Abelian groups

该研究探讨周期阿贝尔群的自同态环与自同构群的初等等价问题，给出了自同态环初等等价的充要条件，并针对特定周期阿贝尔群刻画了自同构群初等等价的等价条件。

输出中的平移长度（F_n）

Translation lengths in Out(F_n)

我们证明了$Out(F_n)$中所有的无限阶元素都有正的平移长度，而且它们都有界远离零。结果包括一个新的证明，可解子群的$Out(F_n)$是可解生成的和虚拟阿贝尔和新的结果，这样的子群是拟凸的。

直角Coxeter群与有限阿贝尔群的图积的初等等价

Elementary equivalence of right-angled Coxeter groups and graph products of finite abelian groups

证明了有限阿贝尔群的图积初等等价当且仅当它们是$\exists\forall$ -等价的当且仅当它们是同构的.特别地，两个直角Coxeter群初等等价当且仅当它们同构。

Coarse models of homogeneous spaces and translations-like actions

For finitely generated groups $G$ and $H$ equipped with word metrics, a translation-like action of $H$ on $G$ is a free action where each element of $H$ moves elements of $G$ a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group $\mathbf{G}$ that is not isogenous to $\mathrm{SL}(2,\mathbb{R})$ admit translation-like actions by $\mathbb{Z}^2$. This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radi…

The finiteness theorem for invariants of a finite group (translation of Emmy Noether's "Der Endlichkeitsatz der Invarianten endlicher Gruppen")

A translation of Emmy Noether's paper "Der Endlichkeitsatz der Invarianten endlicher Gruppen" (Mathematische Annalen, vol. 77, 1920, pages 89--92). In Noether's words, the paper gives "an entirely elementary finiteness proof---using only the theory of symmetric functions---for the invariants of a finite group, which at once supplies an actual statement of a complete system of invariants while the usual proof using the Hilbert basis theorem is only an existence proof."

Garside群是强平移离散群

Garside groups are strongly translation discrete

Garside群是有限型的辫群和Artin群的推广，它被定义为Garside幺半群的分数群.证明了Garside幺半群的半直积是Garside幺半群.利用无限循环群$\mathbb Z$的半直积$\mathbb Z\ltimes G^n$和Garside群$G$的笛卡儿积$G^n$，研究了$G$中元素的根和幂的性质.主要结果是当生成集是简单元素的集合时，估计了Garside群中元素的幂的最小字长的增长。一个直接的应用是Garside群中的平移数的集合是离散的。对Gersten和Short关于Garside群的问题给出了肯定的回答。最初的问题是针对双自动组的。证明了在Garside群$G$中的求根问题可以归结为$\mathbb Z\ltimes G^n$中的共轭问题。利用Garside群中共轭问题的算法，证明了对于任何Garside群，其根的抽取问题都是可判定的。