Related papers: The topology on the space of left orderings of a g…

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…

A natural topology on the space of left orderings of an arbitrary semi-group is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a…

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…

A natural topology on the set of left orderings on free abelian groups and free groups $F_n$, $n>1$ has studied in [1]. It has been proven already that in the abelian case the resulted topological space is a Cantor set. There was a…

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…

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 $SOL=\Z^2\rtimes_T\Z$.

We show how to use topological ideas, such as compactness, to establish orderability properties of infinite groups. A new application is to provide a left-ordering for the group of PL homeomorphisms of a connected surface with boundary…

For any left orderable group G, we recall from work of McCleary that isolated points in the space of left orderings correspond to basic elements in the free lattice ordered group over G. We then establish a new connection between the…

We show that no left-ordering on a free product of (left-orderable) groups is isolated. In particular, we show that the space of left-orderings of free product of finitely generated groups is homeomorphic to the Cantor set. With the same…

In Chapter 1 we give the basic background and notations. We also give a new characterization of the Conrad property for orderings. In Chapter 2, we use the new characterization of the Conradian property to give a classification of groups…

In this note we show that if $G$ is a countably infinite abelian group such that $nG=0$ for some integer $n$, then the only locally minimal group topology on $G$ is the discrete one.

We prove a few basic facts about the space of bi-invariant (or left-invariant) total order relations on a torsion-free, nonabelian, nilpotent group G. For instance, we show that the space of bi-invariant orders has no isolated points (so it…

We classify $C$-orderable groups admitting only finitely many $C$-orderings. We show that if a $C$-orderable group has infinitely many $C$-orderings, then it actually has uncountably many $C$-orderings, and none of these is isolated in the…

In this paper we define countable-configuration of groups and prove that two Hopfian groups with the same set of countable-configurations are isomorphic and vice versa. We also study the countable paradoxical decomposition of groups. It is…

In this work we exhibit flexibility phenomena for some (countable) groups acting by order preserving homeomorphisms of the line. More precisely, we show that if a left orderable group admits an amalgam decomposition of the form…

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…

A left order on a magma (e.g., semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this…

We study the space of ends of groups. For a finitely generated group, this is a Cantor space as soon as it is infinite. In contrast, we show that for infinitely generated countable groups, it exhibits several behaviors. For instance, we…

It is a classical theorem that the free product of ordered groups is orderable. In this note we show that, using a method of G. Bergman, an ordering of the free product can be constructed in a functorial manner, in the category of ordered…

Let $G$ be a group. We can topologize the spaces of left-orderings $LO(G)$ and bi-orderings $O(G)$ of $G$ with the product topology. These spaces may or may not have isolated points. It is known that $LO(F_2)$ has no isolated points, where…