The Amalgamated product of free groups and residual solvability

Residually finite non linear hyperbolic groups

We exhibit the first examples of residually finite non-linear Gromov hyperbolic groups. Our examples are constructed as amalgamated products of torsion-free cocompact lattices in the rank 1 Lie group $\mathrm{Sp}(d,1)$, $d\geq 2$ along maximal cyclic subgroups.

On the residual solvability of generalized free products of solvable groups

In this paper, we study the residual solvability of the generalized free product of solvable groups.

融合积的 pro-p 完备化中的剩余轮廓与有界生成

Residual profiles and bounded generation in pro-$p$ completions of amalgamated products

本文在顶点群有界生成的前提下，分类融合积 pro-p 完备化的有界生成情形，证明其极为罕见并给出反例说明单素数有界生成不蕴含全 profinite 群有界生成。

有理单位群的 amalgam 与残余 nilpotence

Amalgams of rational unipotent groups and residual nilpotence

研究有理单位群的 amalgam 在残余 nilpotence 中的条件，并通过 Cartan 矩阵特征化高维 amalgam 的残余 nilpotence，给出有理 Kac-Moody 群的正规形式。

Some Residually Solvable One-Relator Groups

This communication records some observations made in the course of studying one-relator groups from the point of view of residual solvability. As a contribution to clas- sification efforts we single out some relator types that render the corresponding one-relator groups residually solvable.

Central amalgamation of groups and the RFD property

It is an old and challenging topic to investigate for which discrete groups G the full group C*-algebra C*(G) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [CS19] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows…

Euler characteristics on virtually free products

We define Euler characteristics on classes of residually finite and virtually torsion free groups and we show that they satisfy certain formulas in the case of amalgamated free products and HNN extensions over finite subgroups. These forumlas are obtained from a general result which applies to the rank gradient and the first $L^2$-Betti number of a finitely generated group.

Groups satisfying Kaplansky's stable finiteness conjecture

We prove that every {finitely generated residually finite}-by-sofic group satisfies Kaplansky's direct and stable finiteness conjectures with respect to all noetherian rings. We use this result to provide countably many new examples of finitely presented non-LEA groups, for which soficity is still undecided, satisfying these two conjectures. Deligne's famous example of a non residually finite group is among our examples, along with the families of amalgamated free products SL_n(Z[1/p])*_{F_r}SL_n(Z[1/p]) and HNN extensions SL_n(Z[1/p])*_{F_r}, where p>2 is a prime, n>2 and F_r is a free group …

The cyclic subgroup separability of certain generalized free products of two groups

Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups.

斜括号中的理想与可解性

Ideals and Solvability in Skew Braces

研究有限斜括号乘法群幂零性对其理想结构和可解性的影响，证明相关结论，探讨双边斜括号可解性，得出有限双边斜括号可解性与加或乘法群可解性的关系及无限情况下的相关结论。

Classes of Groups Generalizing a Theorem of Benjamin Baumslag

In [BB] Benjamin Baumslag proved that being fully residually free is equivalent to being residually free and commutative transitive (CT). Gaglione and Spellman [GS] and Remeslennikov [Re] showed that this is also equivalent to being universally free, that is, having the same universal theory as the class of nonabelian free groups. This result is one of the cornerstones of the proof of the Tarksi problems. In this paper we extend the class of groups for which Benjamin Baumslag's theorem is true, that is we consider classes of groups $\X$ for which being fully residually $\X$ is equivalent to be…

CSA groups and separated free constructions

A group $G$ is said to be a {\it CSA}-group if all maximal abelian subgroups of $G$ are malnormal. The class of CSA groups is of interest because it contains torsion-free hyperbolic groups, groups acting freely on $Λ$-trees and groups with the same existential theory as free groups. CSA groups are also very closely related to the study of residually free groups and tensor completions. In this paper we investigate which free constructions (amalgamated products and HNN extensions) over CSA groups are again CSA. The results are applied, in particular, to show that a torsion-free one-relator group…

Reidemeister类、圈积与可解性

Reidemeister classes, wreath products and solvability

本文研究了限制圈积中的Reidemeister类，在自同构阶数和Reidemeister类数目有限的条件下证明了猜想TBFT_f，并证明了满足该条件的有限生成剩余有限群具有可解有限性（猜想R）。

Automaton groups and complete square complexes

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and CAT(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three squares, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete VH complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually…

链环的基本 $n$-quandle 是剩余有限的

Fundamental $n$-quandles of links are residually finite

本文研究了quandle的剩余有限性与子quandle可分性，证明了3-球面中任意定向链环的基本$n$-quandle（$n\ge2$）是剩余有限的，并给出了相关有限性质的一般性结果及多个满足该性质的quandle族。

The conjugacy problem for relatively hyperbolic groups

Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this assertion. We conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups.

圈积与半直积的几何

The geometry of wreath and semi-direct products

本文研究了圈积与半直积在陪集几何上的扩展，证明了这些操作保持旗传递性、剩余连通性和薄性，并应用于构造几乎单群的正则多胞体。

从群运算到几何结构：陪集几何中的融合、HNN-扩张与扭转

From Group Operations to Geometric Structures: Amalgamations, HNN-Extensions, and Twisting in Coset Geometries

本文将融合积、HNN-扩张等经典群论构造扩展至陪集几何，提出一种在保留旗传递性等性质下粘合关联几何的通用框架，并据此分析了 Shephard 群族，揭示了陪集几何的 Bass-Serre 理论。