A closer look at the non-Hopfianness of $BS(2,3)$

关于秩 $2$ 单代数群中的极小满射子群

On minimal epimorphic subgroups in simple algebraic groups of rank $2$

本文研究了秩 $2$ 单代数群中的极小满射子群问题，证明了在代数闭域上其闭满射子群的极小维数为 $3$。

Indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2 with non-abelian permutation group

We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level~2. In the first step we present a construction of some family of such solutions and in the second step we prove that every indecomposable involutive solution of the Yang-Baxter equation with multipermutation level~2 is a homomorphic image of a solution previously constructed. Analyzing this epimorphism, we are able to obtain all such solutions up to isomorphism and enumerate these of small sizes.

Generalized E-Rings

A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End(R_Z) of the additive group R_Z, taking any r in R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this case R_Z is called an E-group. Obvious examples of E-rings are subrings of Q. However there is a proper class of examples constructed recently. E-rings come up naturally in various topics of algebra. This also led to a generalization: an abelian group G is an E-group if there is an epimorphism from G onto the additive group of End(G). If G is torsion-free of …

Tits alternatives for graph products

We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the …

Large groups and their periodic quotients

We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non--cyclic free group, and $g$ is an element of $H$, then the quotient of $G$ by the normal subgroup generated by $g^n$ is large for all but finitely many $n\in \mathbb Z$. In the second part of this note we use similar methods to show that for every infinite sequence of primes $(p_1, p_2, ...)$, there exists an infinite finitely generated periodic group $Q$ with descending normal series $Q=Q_0\rhd Q_1\rhd ... $, such t…

非平面端点始终是连续不可忘的

Non-planar ends are continuously unforgettable

该研究证明了特定映射类群子群之间的连续满射由同胚诱导，并推导出这些群是霍普夫拓扑群的结论。

Coarse non-amenability and covers with small eigenvalues

Given a closed Riemannian manifold M and a (virtual) epimorphism from the fundamental group of M onto a free group of rank 2, we construct a tower of finite sheeted regular covers {M_n}_{n=0}^{\infty} of M such that the first non-zero eigenvalues λ_1(M_n) of the Laplacian converge to zero as n tends to infinity. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where the fundamental group of M is supposed to surject onto an amenable group.

杨-巴克斯特方程的不可分解非退化2-置换解

Indecomposable non-degenerate 2-permutational solutions of the Yang-Baxter equation

该研究完整刻画了杨-巴克斯特方程2-置换水平的不可分解非退化解，通过构造通用解同态像及基于群乘积的解族，提供了找到此类解的工具，并指出其自同构群正则。

每个非平凡纽结群都是完全剩余完美群

Every non-trivial knot group is fully residually perfect

本文证明每个非平凡纽结群均为完全剩余完美闭3-流形群，拓展了非平凡纽结群的完全剩余性质相关结论。

Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements displaced by any non-identity automorphism. In this paper we confirm Tom Tucker's "Infinite Motion Conjecture" that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of a sequence of finite permutation groups with disjoint domains, viewed as a permutation gr…

Notes on non-archimedean topological groups

We show that the Heisenberg type group $H_X=(\Bbb{Z}_2 \oplus V) \leftthreetimes V^{\ast}$, with the discrete Boolean group $V:=C(X,\Z_2)$, canonically defined by any Stone space $X$, is always minimal. That is, $H_X$ does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean $G$ there exists a (resp., locally compact) non-archimedean minimal group $M$ such that $G$ is a group retract of $M.$ For discrete groups $G$ the latter was proved by S. Dierolf and U. Schwanengel. We unify some old and new characteriza…

群的完备嵌入

Complete Embeddings of Groups

本文证明每个可数群都可嵌入到一个有限生成、hopfian 且完备的群中，并借助非对称非 Haken 闭双曲 3-流形完成构造，同时保持有限表示与有限分类空间性质。

短层次超几何群II：商和Artin群的Hopf性质

Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups

本文证明大多数大类型和超几何类型的Artin群是Hopf群，并展示大部分五孔球面映射群的商是层次超几何的。