Non-$\forall$-homogeneity in free groups

Friedman的$ \mathsf{WD} $不是无参数序列的

Friedman's $ \mathsf{WD} $ is not parameter-free sequential

本文回答了Visser关于弱集合论$ \mathsf{WD} $是否为无参数序列的问题，通过力迫法构造模型证明了$ \mathsf{WD} $及其外延性扩展均不是无参数序列的。

不可数相对自由群的构造原理与非齐次性

The construction principle and non homogeneity of uncountable relatively free groups

本文针对Belegradek提出的剩余有限性假设是否必要的问题，利用无穷逻辑方法给出否定回答，证明含非可解群的簇中不可数V-自由群非ℵ₁-齐次，推广了相关结果。

Elementary subgroups of virtually free groups

We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors. Moreover, we give an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $\exists\forall\exists$-elementary. We also prove that every elementary embedding of an equationally noetherian group into itself is an automorphism.

关于自由群因子的一阶择一性问题

On the First-Order Free Group Factor Alternative

本文研究自由群因子是否共享同一一阶理论，为其共同一阶基本群建立三分法，并证明二分法在 $\forall \exists$-理论层面成立。

Coxeter群与分裂晶体群的同质性

Homogeneity in Coxeter groups and split crystallographic groups

本文研究了Coxeter群和晶体群的同质性，证明了某些双曲群和Coxeter群的同质性，并构造了非同质的分裂晶体群。

Implicit function theorem over free groups

We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the Implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lifting solutions of equations into generic points. From the model theoretic view-point we claim the existence of simple Skolem functions for particular $\forall\exists$-formulas over free groups. Proving th…

On expansions of non-abelian free groups by cosets of a finite index subgroup

Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $q\in Q$ to the language of groups. Using a slight generalization of some of the techniques involved in Zlil Sela's solution to Tarskiś problem on the elementary theory of non-abelian free groups, we provide a few basic results on the validity of first order entences in the $L_Q$-expansion of $F$ in which every $P_q$ is interpreted as the preimage of $q$ in $F$. In particular we prove an analogous result to Sela's generalization of Merzlyakov…

Fixed point free homeomorphisms of the complex plane

Our purpose in this article is to prove that the group $H({\bf C})$ of homeomorphisms of the complex plane ${\bf C}$ is a metric group equipped with the metric induced by uniform convergence of homeomorphisms and their inverses on compacts and the set $$\left\{ h \in H({\bf C}) : ( \forall z \in {\bf C} )( h(z) \neq z ) \right\} $$ of fixed point free homeomorphisms of the complex plane is a conjugacy invariant dense $G_{δ}$ subset of $H({\bf C})$.

Virtually free groups are almost homogeneous

Free groups are known to be homogeneous, meaning that finite tuples of elements which satisfy the same first-order properties are in the same orbit under the action of the automorphism group. We show that virtually free groups have a slightly weaker property, which we call uniform almost-homogeneity: the set of $k$-tuples which satisfy the same first-order properties as a given $k$-tuple $\mathbf{u}$ is the union of a finite number of $\mathrm{Aut}(G)$-orbits, and this number is bounded independently from $\mathbf{u}$ and $k$. Moreover, we prove that there exists a virtually free group which i…

Formal solutions and the first-order theory of acylindrically hyperbolic groups

We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique maximal finite normal subgroup of $G$, then $G$ and the HNN extension $G\dot{\ast}_{E(G)}$, which is simply the free product $G\ast\mathbb{Z}$ when $E(G)$ is trivial, have the same $\forall\exists$-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez González: acylindrically hyperbolic groups have trivi…

自由格的初等性质 II：全称理论的可判定性

Elementary properties of free lattices II: Decidability of the universal theory

本文研究了自由格的模型论，证明了无限自由格的全称理论可判定，并展示了有限生成自由格的正向可区分性以及投影格的理论性质。

有限有界同质性的内外翻转

Finitely Bounded Homogeneity Turned Inside-Out

本文将有限有界类的合并性质判定归约联系于归约包含问题，证明其2NEXPTIME困难，并在可计算有限有界Ramsey扩张假设下给出可判定性，同时证明可齐次化检验不可判定。

$\aleph_1$-free abelian非阿基米德波兰群

$\aleph_1$-free abelian non-Archimedean Polish groups

本文研究了$\aleph_1$-free交换非阿基米德波兰群，证明了存在连续多个可分离的此类群，且四个性质在闭子群空间中为完全分析集。

齐次无等腰空间

Homogeneous isosceles-free spaces

本文刻画了齐次无等腰空间为二元域上带内射范数的向量空间，并利用无等腰分解给出齐次有限度量空间最大距离数的界。