Yibei Li
We apply results proved in [Li19] to the linear order expansions of non-trivial free homogeneous structures and the universal n-linear order for $n\geq 2$, and prove the simplicity of their automorphism groups.
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An infinite linearly ordered set (S,<=) is called doubly homogeneous if its automorphism group A(S) acts 2-transitively on it. We show that any group G arises as outer automorphism group G cong Out(A(S)) of the automorphism group A(S), for some doubly homogeneous chain (S,<=).
We characterize the groups isomorphic to full automorphism groups of ordered abelian groups. The result will follow from classical theorems on ordered groups adding an argument from proofs used to realize rings as endomorphism rings of abelian groups.
We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countable linearly ordered set $Ω$, there are uncountably many maximal subgroups of $\operatorname{End}(\mathbb{Q},{\leq})$ isomorphic to the automorphism group of $Ω$. We characterise those subsets $X$ of $\mathbb{Q}$ that arise as a retract in $(\mathbb{Q},{\leq})$ in terms of topological information concerning $X$. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal…
This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager $2$-dimensional diagonal conjugacy class. We also provide general conditions im…
Groups with infinite linearly ordered products
本文提出无限线性有序乘积的形式体系，用于定义超序列等幂级数群中的无限复合，并证明此类群可表示为有序阿贝尔群的半直积。
Ordered bases, order-preserving automorphisms and bi-orderable link groups
本文提出了一种准则，证明自由群在特定自同构下可双可排序，并应用该结果证明魔法流形的基本群可双可排序。
We give a criterium when a linearly ordered topological semilattice is $H$-closed. We also prove that any linearly ordered $H$-closed topological semilattice is absolutely $H$-closed and we show that every linearly ordered semilattice is a dense subsemilattice of an $H$-closed topological semilattice.
For a linearly ordered group $G$ let us define a subset $A\subseteq G$ to be a \emph{shift-set} if for any $x,y,z\in A$ with $y < x$ we get $x\cdot y^{-1}\cdot z\in A$. We describe the natural partial order and solutions of equations on the semigroup $\mathscr{B}(A)$ of shifts of positive cones of $A$. We study topologizations of the semigroup $\mathscr{B}(A)$. In particular, we show that for an arbitrary countable linearly ordered group $G$ and a non-empty shift-set $A$ of $G$ every Baire shift-continuous $T_1$-topology $τ$ on $\mathscr{B}(A)$ is discrete. Also we prove that for an arbitrary …
A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. The groups $\operatorname{Aut}(A)$ of such posets $A$ have a natural topology in which $\operatorname{Aut}(A)$ are Polish topological groups. We consider the problem whether $\operatorname{Aut}(A)$ contains a dense free subgroup of two generators. We show that if $A$ is ultrahomogeneous, then $\operatorname{Aut}(A)$ contains such subgroup. Moreover, we characterize whose countable ultrahomogeneous posets $A$ such that for each natural $m$, the set of all cyclically dense …
In the paper we study inverse semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$ which are generated by partial monotone injective translations of a positive cone of a linearly ordered group $G$. We describe Green's relations on the semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$, their bands and show that they are simple, and moreover the semigroups $\mathscr{B}(G)$ and $\mathscr{B}^+(G)$ are bisimple. We show that for a commutative linearly ordered group $G$ all non-trivial congruences …
We present a normal form for outer automorphisms $ϕ$ of a non-abelian free group $F_N$ which grow quadratically (measured through the maximal growth of conjugacy classes in $F_N$ under iteration of $ϕ$). In analogy to the known normal form for linearly growing automorphisms as efficient Dehn twist, our normal form for $ϕ$ is given in terms of a 2-level Dehn twist on a graph-of-groups $\cal{G}$ with $π_1 {\cal{G}} \cong F_N$, where a conjugacy class of $F_N$ grows at most linearly if and only if it is contained in a vertex group of $\cal{G}$. Our proof is based on earlier work of the second aut…
We investigate correspondences between extreme amenability and amenability of automorphism groups of Fraïssé-Hrushovski generic structures that are obtained from smooth classes, and their Ramsey type properties of their smooth classes, similar to Kechris, Pestov and Todorcevic, and Tatch Moore. In particular, we focus on some Fraïssé-Hrushovski generic structures that are obtained from pre-dimension functions. Using these correspondences, we prove that automorphism groups of ordered Hrushovski generic graphs are not extremely amenable in both cases of collapsed and uncollapsed. Moreover, we pr…
We show that the level sets of automorphisms of free groups with respect to the Lipschitz metric are connected as subsets of Culler-Vogtmann space. In fact we prove our result in a more general setting of deformation spaces. As applications, we give metric solutions of the conjugacy problem for irreducible automorphisms and the detection of reducibility. We additionally prove technical results that may be of independent interest --- such as the fact that the set of displacements is well ordered.
We call the family of free-by-cyclic groups defined by $G = \left< a, t, b_1, b_2, \ldots b_k \mid at = ta, b_1^{-1}tb_1 = a^{n_1}t, \ldots b_k^{-1}tb_k = a^{n_k}t \right>$ for $n_1, n_2, \ldots n_k \in \mathbb Z$ linearly mismatched since the automorphisms used to define the HNN extensions grow linearly at different rates. Using techniques from Elder's thesis, namely words with a parallel stable letter structure, we prove that linearly mismatched free-by-cyclic groups are asynchronously automatic, and thus they have a solvable word problem.
We consider a finite, connected and simple graph $Γ$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(Γ)$, the local action $G_x^{Γ(x)}$ is the action of $\mathrm{Sym}(n)$ on ordered pairs, we show that the group $G_x^{[3]}$, the pointwise stabiliser of a ball of radius three around $x$, is trivial.
It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is red…
Ordered groups of formal series, and a conjugacy problem
本文研究形式级数有序域上的复合运算构成群的条件，证明超越级数域和超级数域满足条件，并进一步给出构造恰好具有三个共轭类的线性有序群的条件，解决了该开放存在性问题。
We classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in terms of periodic orbits of a birational action of an infinite Coxeter group on ordered point sets in the projective plane modulo projective equivalence. We also outline the classification of non-rational surfaces with large automorphism groups.
On ordered groups of regular growth rates
本文引入增长阶群，涵盖形式级数与o-极小结构中的无穷大芽群，研究其代数结构并给出构造方法，证明在分层条件下此类群为增长阶群。
Centralisers of linear growth automorphisms of free groups
本文研究自由群 Out(F_n) 中线性增长元素的中心化子，通过有限指数子群到等变 McCool 群直积的映射结构，证明其具有 VF 性质并因而有限展示。
Idempotents, automorphism groups, and commutator widths of quandle algebras
本文研究Quandle代数，证明特征不为2的整环上有序交换Quandle的Quandle代数无非平凡幂等元，计算两类Quandle代数的自同构群，通过搜索得到换位子宽度为2的首批Quandle代数例子。
Let $\mathbb{A} = (A, \cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that $\mathbb{A}$ is linearly orderable if there exists a total order $\le$ on $A$ such that $xz < yz$ and $zx < zy$ for all $x,y,z \in A$ with $x < y$. In particular, we find that if $S$ is a finite subset of $A$ generating a non-abelian subsemigroup of $\mathbb{A}$, then $|S^2| \ge 3|S|-2$. On the road to this goal, we also prove a number of…
We study, from a constructive computational point of view, the techniques used to solve the conjugacy problem in the "generic" lattice-ordered group Aut(R) of order automorphisms of the real line. We use these techniques in order to show that for each choice of parameters f,g in Aut(R), the equation xfx=g is effectively solvable in Aut(R).
Let K be a cyclic Galois extension of degree f over k and T a generator of the Galois group. For any v=(v_1,... , v_m)\in K^m such that v is linearly independent over k, and any 0< d < m the Gabidulin-like code C(v, T , d) is a maximum rank distance code in the space of f times m matrices over k of dimension fd. This construction unifies the ones available in the literature. We characterise the K-linear codes that are Gabidulin-like codes and determine their rank-metric automorphism group.
A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on poi…
Free products of bi-orderable groups
本文证明了双序群自由积不存在孤立双序，并证明$Aut(F_2)$在$F_2$上的自然作用无稠密轨道。
Orbits of automorphism groups of partially ordered sets are not necessarily congruence classes, i.e. images of an order homomorphism. Based on so-called orbit categories a framework of factorisations and unfoldings is developed that preserves the antisymmetry of the order Relation. Finally some suggestions are given, how the orbit categories can be represented by simple directed and annotated graphs and annotated binary relations. These relations are reflexive, and, in many cases, they can be chosen to be antisymmetric. From these constructions arise different suggestions for fundamental syste…
McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $σ,τ\in \mathrm{Mod}(S)$ there is some power $N$ such that the group $\langle σ^N,τ^N\rangle$ is either free of rank two or abelian, and gives a geometric criterion for the dichotomy. The analogous statement is false in linear groups, and unresolved for outer automorphisms of a free group. Several analogs are known for exponentially growing outer automorphisms satisfying various technical hypothesis. In this article we prove an analogous statement when $σ$ and $τ$ are linearly …
Proper partial linear spaces affording imprimitive rank 3 automorphism groups
本研究针对真部分线性空间的对称分类问题，首次完成了半本原非本原秩3真部分线性空间的分类，涵盖本原传递、拟本原群情形，构造了多族无限实例及10个单独实例，多数为文献中新结果。
Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every …
A masure (a.k.a affine ordered hovel) I is a generalization of the Bruhat-Tits building that is associated to a split Kac-Moody group G over a non-archimedean local field. This is a union of affine spaces called apartments. When G is a reductive group, I is a building and there is a G-invariant distance inducing a norm on each apartment. In this paper, we study distances on I inducing the affine topology on each apartment. We show that some properties (completeness, local compactness, ...) cannot be satisfyed when G is not reductive. Nevertheless, we construct distances such that each element …
Unipotent linear suspensions of free groups
受自由群外自同构共轭问题驱动，本文发展了由幂单线性增长自同构产生的 free-by-cyclic 群的算法理论，计算了其规范分裂与自同构群，证明了有效凝聚性，解决了混合 Whitehead 问题，并最终应用于解决该类自由群外自同构的共轭问题。
Let $(Ω, \leq)$ be a totally ordered set. We prove that if Aut$(Ω,\leq)$ is transitive and satisfies the same first-order sentences as the automorphism group of the real line (in the language of groups) then $Ω$ and and the real line are isomorphic ordered sets. This improvement of a theorem of Gurevich and Holland is obtained as a consequence of a study of centralizers associated with certain transitive subgroups of Aut$(Ω,\leq)$.
Let p be a prime, G a locally finite p-group, K a commutative ring of characteristic p. The anti-automorphism g\mapsto g\m1 of G extends linearly to an anti-automorphism a\mapsto a^* of KG. An element a of KG is called symmetric if a^*=a. In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group.
We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard-Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial…
This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the…
Automorphism group of the graph $A(n,k,r)$
本文刻画了排列图 $A(n,k,r)$ 在 $1\leq r=k\leq n$ 及 $r=2<k=n$ 两种情形下的全自同构群，解决了前人提出的两个公开问题特例，并提出新猜想。
We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper classifies those lines that exhibit almost simple symmetries. We introduce a general recipe involving Schur covers to recover doubly transitive lines from their automorphism group. Combining our results with recent work on the affine case by Dempwolff and Kantor, we deduce a classification of all linearly dependent doubly transitive lines in real or complex space.
We address the "sums of dilates problem" by looking for non-trivial lower bounds on sumsets of the form $k \cdot X + l \cdot X$, where $k$ and $l$ are non-zero integers and $X$ is a subset of a possibly non-abelian group $G$ (written additively). In particular, we investigate the extension of some results so far known only for the integers to the context of torsion-free or linearly orderable groups, either abelian or not.
A hovel is a generalization of the Bruhat-Tits building that is associated to an almost split Kac-Moody group G over a non-Archimedean local field. In particular, G acts strongly transitively on its corresponding hovel $Δ$ as well as on the building at infinity of $Δ$, which is the twin building associated to G. In this paper we study strongly transitive actions of groups that act on affine ordered hovels $Δ$ and give necessary and sufficient conditions such that the strong transitivity of the action on $Δ$ is equivalent to the strong transitivity of the action of the group on its building at …
Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $π$ acts $2$-transitively on the points of $π$, then $π$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{PΓL}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a \emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the \emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-tra…
On average orders of automorphism groups of bilinear maps over finite fields
本文研究有限域上双线性映射在一般线性群作用下的同构类数量渐近界，并将其应用于 Frattini 类 2 的 p-群计数、平凡自同构群矩阵空间采样及立方零交换代数同构类估计。
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