Embeddings and chains of free groups

On the dimension of matrix embeddings of torsion-free nilpotent groups

Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices $UT_N(Z)$ for some $N$. In 2006, Nickel proposed an algorithm to calculate such embeddings. In this work, we show that if $UT_n(Z)$ is embedded into $UT_N(Z)$ using Nickel's algorithm, then $N \geq 2^{n/2 -2}$ if the standard ordering of the Mal'cev basis (as in Nickel's original paper) is used. In particular, we establish an exponential worst-case running time of Nickel's algorithm. On the other hand, we also prove a general exponenti…

递归群显式嵌入的辅助自由构造

Auxiliary free constructions for explicit embeddings of recursive groups

本文提出一种基于HNN-扩张和融合自由积的辅助自由构造，推广了Higman的结构，简化了递归群嵌入方法，并用于在小秩自由群中构造无限生成的良性子群。

Effective quasimorphisms on free chains

We find homogeneous counting quasimorphisms that are effective at seeing chains in a free group F. As corollary, we derive that if a group G has an index-d free subgroup, then every element g in G either has stable commutator length at least 1/8d or some power of g is conjugate to its inverse. We also show that for a finitely-generated free group F, there is a countable basis for the real vector space of homogeneous quasimorphisms on F.

将有限生成自由-由循环群嵌入到有限生成自由-由循环群中

Embedding finitely generated free-by-cyclic groups in {finitely generated free}-by-cyclic groups

本文改进了Feighn-Handel关于自由群映射筒子群的结果，证明有限生成自由-由循环群可嵌入到有限生成自由-由循环群中，并在超几何情况下作为准凸子群嵌入。

Preserving torsion orders when embedding into groups with `small' finite presentations

We give a complete survey of a construction by Boone and Collins for embedding any finitely presented group into one with $8$ generators and $26$ relations. We show that this embedding preserves the set of orders of torsion elements, and in particular torsion-freeness. We combine this with the independent results of Belegradek and Chiodo to prove that there is an $8$-generator $26$-relator universal finitely presented torsion-free group (one into which all finitely presented torsion-free groups embed).

性质(LR)与虚自由群的嵌入定理

Property (LR) and an embedding theorem for virtually free groups

本文证明了每个虚自由群G具有Long和Reid提出的性质(LR)，即每个有限生成子群都是有限指数子群的 retract。主要方法是证明每个可数虚自由群可嵌入到有限群的双群中。

秩为2的自由群中使用通用词的嵌入

Embeddings using universal words in the free group of rank 2

本文提出一种将可数群G=<A|R>嵌入2-生成元群T的简单方法，使生成元像明确且T的定义关系可自动推导，并展示了具体应用与递归群嵌入的参考。

Trivial cup products in bounded cohomology of the free group via aligned chains

We prove that the cup product of $Δ$-decomposable quasimorphisms, Brooks quasimorphisms or Rolli quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial.

The Hahn embedding theorem for a class of residuated semigroups

Hahn's embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn's theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial sublex product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess o…

Elementary embeddings in torsion-free hyperbolic groups

We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We deduce as a corollary that subgroups elementarily embedded in finitely generated free groups are free factors.

自由拟凸子群的严格上升链

Ascending Chains of Free Quasiconvex Subgroups

本文证明了双曲群中不存在常秩自由拟凸子群的严格上升链，这一结果深化了对双曲群子群结构的理解。

稳健的虚拟自由群的准等距嵌入

Robust quasi-isometric embeddings of virtually free groups

论文研究了非元素自由群在GL_n(k)中的稳健准等距嵌入，展示了非Anosov表示的示例，并探讨了自由半群的嵌入特性。

Embedding problems and open subgroups

We study the properties of the fundamental group of an affine curve over an algebraically closed field of characteristic $p$, from the point of view of embedding problems. In characteristic zero, the fundamental group is free, but in characteristic $p$ it is not even $ω$-free. In this paper we show that it is "almost $ω$-free," in the sense that each finite embedding problem has a proper solution when restricted to some open subgroup. We also prove that embedding problems can always be properly solved over the given curve if suitably many additional branch points are allowed, in locations that…

自由群与自由群因子的测度等价嵌入

Measure equivalence embeddings of free groups and free group factors

本文证明自由群$\mathbb{F}_2$可嵌入任意非 amenable lcsc 群，据此推导非 amenable lcsc 群的遍历作用性质，并为$II_1$因子引入测度等价概念，证明其非 amenable 性与自由群因子嵌入的等价性及性质稳定性。