Peter A. Linnell
Let G be a group and let O_G denote the set of left orderings on G. Then O_G can be topologized in a natural way, and we shall study this topology to show that O_G can never be countably infinite. This paper retrieves correct parts of the withdrawn paper arXiv:math/0607470.
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The space of Conradian left-preorders
本文研究 Conradian 左预序空间，证明其必为有限或不可数，并给出空间有限性的等价条件及 Conradian 左预序的若干刻画。
Work of Linnell shows that the space of left-orderings of a group is either finite or uncountable, and in the case that the space is finite, the isomorphism type of the group is known---it is what is known as a Tararin group. By defining semiconjugacy of circular orderings in a general setting (that is, for arbitrary circular orderings of groups that may not act on $S^1$), we can view the subspace of left-orderings of any group as a single semiconjugacy class of circular orderings. Taking this perspective, we generalize the result of Linnell, to show that every semiconjugacy class of circular …
Let G be a left orderable group and LO(G) the space of all left orderings. We investigate the circumstances under which a left ordering < of G can correspond to an isolated point in LO(G), in particular we extend known results to cover the case of uncountable groups. With minor technical restrictions on the group G, we also find that no dense left ordering is isolated in LO(G), and that the closure of the set of all dense left orderings of G yields a dense G-delta set within a Cantor set of left orderings in LO(G). Lastly, we show that certain conditions on a discrete left ordering of G can gu…
We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along the way, we define and investigate the space of locally invariant orderings of a group, the natural group actions on this space, and their relationship to the space of left-orderings.
We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semi-direct product decomposition. This allows us to also show that the space of circular orders of any group is either finite or uncountable. As a special case and first step, we show that the space of circular orderings of an infinite Abelian group has no isolated points, hence is homeomorphic to a cantor set.
Classification of finite left-simple skew braces
本文给出了有限左单斜括号的完整分类，证明了非平凡有限斜括号无非平凡真左理想当且仅当其为素数阶平凡斜括号或与有限非阿贝尔单群相关的几乎平凡斜括号。
For an uncountable cardinal τand a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns:s\in S}|\ge τfor all integers n\ge 1; (ii) there exists a group homomorphism π:G\to T^{2^τ} such that π(S) is dense in T^{2^τ}. Moreover, if |G|\le 2^{2^τ}, then the following item can be added to this list: (iii) there exists an isomorphism π:G\to G' between G and a subgroup G' of T^{2^τ} such that π(S) is dense in T^{2^τ}. We prove that the following conditions are equivalent for an uncountable subset S of an abelian group G that is either (almost) torsion-free or divisible: (…
Spaces of left-preorders on free products
本文利用动力系统方法研究自由积上的左预序空间，证明其中无孤立元素，并在有限生成情形下证明该空间为空或康托集，同时将结果推广到有限Kurosh秩子群并给出相关子群交的生成集。
Every left-invariant ordering of a group is either discrete, meaning there is a least element greater than the identity, or dense. Corresponding to this dichotomy, the spaces of left, Conradian, and bi-orderings of a group are naturally partitioned into two subsets. This note investigates the structure of this partition, specifically the set of dense orderings of a group and its closure within the space of orderings. We show that for bi-orderable groups this closure will always contain the space of Conradian orderings---and often much more. In particular, the closure of the set of dense orderi…
This paper is a continuation of [BaSh], where we studied the behaviour of the abelianization functor under inverse limits. Our main result in [BaSh] was that if $\mathcal{T}$ is a countable directed poset and $G:\mathcal{T}\to\mathcal{G} rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$\mathrm{Ab}({\lim}_{t\in\mathcal{T}}G_t)\to {\lim}_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$ is surjective, and its kernel is a cotorsion group. The abelianization is an example of a left adjoint functor from groups to abelian groups. In this paper we study the behaviour u…
We show that the space of left-orderings of a countable virtually solvable group is either finite or homeomorphic to a Cantor set. We also provide an explicit description of the space of left-orderings of $SOL=\Z^2\rtimes_T\Z$.
The Burns-Hale theorem states that a group G is left-orderable if and only if G is locally projectable onto the class of left-orderable groups. Similar results have appeared in the literature in the case of UPP groups and Conradian left-orderable groups, with proofs using varied techniques in each case. This note presents a streamlined approach to showing that if C is the class of either Conradian left-orderable, left-orderable, or UPP groups, then C contains all groups that are locally projectable onto C; and shows that this streamlined approach works for the class of diffuse groups as well. …
This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It begins with an introduction to orderable groups and their algebraic properties. Many of the algebraic results are proved by topological methods, via consideration of the space of orderings. After a discussion Hölder's theorem and some dynamical aspects of orderable groups, we provide explicit orderings of important groups in topology: free groups and most surface…
Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity $(x*y)*(x*z)=(y*y)*(y*z)$. Using combinatorial properties of the Cayley kernel and the squaring mapping, we prove that a twisted Ward left quasigroup of prime order is either permutational or a quasigroup. Up to isomorphism, all twisted Ward quasigroups $(X,*)$ are obtained by twisting the left division operation in groups (that is, they are of the form $x*y=ψ(x^{-1}y)$ for a group $(X,\cdot)$ and its automor…
We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity. This allows us to reformulate in a common framework former results by Le Donne-Xie, Eskin-Fisher-Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner-Müller-Xie, on quasiisometries of these solvable groups.
Determining skew left braces of size np
本文定义了斜左括号的二重半直积，在特定条件下证明了大小为np的斜左括号的结构，并给出算法与计数公式，进而描述12p大小的斜左括号，证明了一个猜想。
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic unit vector field. From geometrical viewpoint, the field is either parallel or characteristic vector field of a natural almost contact structure on the group.
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