We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the “magic manifold” is bi-orderable, answering a question of Kin and Rolfsen.

Let g be an element of a group G. For a positive integer n, let $R_n(g)$ be the subgroup generated by all commutators $[\ldots [[g,x],x],\ldots ,x]$ over $x\in G$, where x is repeated n times. Similarly, $L_n(g)$ is defined as the subgroup generated by all commutators $[\ldots [[x,g],g],\ldots ,g]$, where $x\in G$ and g is repeated n times. In the literature, there are several results showing that certain properties of groups with small subgroups $R_n(g)$ or $L_n(g)$ are close to those of Engel groups. The present article deals with orderable groups in which, for some $n\geq 1$, the subgroups $R_n(g)$ are polycyclic. Let $h\geq 0$, $n>0$ be integers and G be an orderable group in which $R_n(g)$ is polycyclic with Hirsch length at most h for every $g\in G$. It is proved that there are $(h,n)$-bounded numbers $h^*$ and $c^*$ such that G has a finitely generated normal nilpotent subgroup N with $h(N)\leq h^*$ and $G/N$ nilpotent of class at most $c^*$. The analogue of this theorem for $L_n(g)$ was established in 2018 by Shumyatsky [‘Orderable groups with Engel-like conditions’, J. Algebra 499 (2018), 313–320].

A nontrivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for generalized torsion orders by using the Alexander polynomial.

The group of order-preserving automorphisms of a finitely generated Archimedean ordered group of rank $2$ is either infinite cyclic or trivial according as the ratio in $\mathbb {R}$ of the generators of the subgroup is or is not quadratic over $\mathbb {Q}.$ In the case of an Archimedean ordered group of rank $2$ that is not finitely generated, the group of order-preserving automorphisms is free abelian. Criteria determining the rank of this free abelian group are established.

We examine a cyclic order on the directed edges of a tree whose vertices have cyclically ordered links. We use it to show that a graph of groups with left-cyclically ordered vertex groups and convex left-ordered edge groups is left-cyclically orderable.

The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) there exists a bi-orderable, two-generated computably presented solvable group with undecidable word problem. Both of the groups can be found among two-generated solvable groups of derived length $3$.

(1) [a]nswers a question posed by Downey and Kurtz; (2) answers a question posed by Bludov and Glass in Kourovka Notebook.

One of the technical tools used to obtain the main results is a computational extension of an embedding theorem of B. Neumann that was studied by the author earlier. In this paper we also compliment that result and derive new corollaries that might be of independent interest.

To every dynamical system $(X,\varphi )$ over a totally disconnected compact space, we associate a left-orderable group $T(\varphi )$. It is defined as a group of homeomorphisms of the suspension of $(X,\varphi )$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple and, if it is a subshift, then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system $(X,\varphi )$, the group $T(\varphi )$ does not have infinite subgroups with Kazhdan's property $(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally, if $(X,\varphi )$ has a dense orbit, then the isomorphism type of the group $T(\varphi )$ is a complete invariant of flow equivalence of the pair $\{\varphi , \varphi ^{-1}\}$.

We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another.

As an application, we provide a short proof (modulo Bass–Serre theory) of Vinogradov’s result that the free product of left-orderable groups is left-orderable.

We study 2-generated subgroups $\langle f,g\rangle <\operatorname{Homeo}^{+}(I)$ such that $\langle f^{2},g^{2}\rangle$ is isomorphic to Thompson’s group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $C^{2}$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $F$ as part of a general phenomenon among subgroups of $\operatorname{Homeo}^{+}(I)$.

We use quasi-orders to describe the structure of C-groups. We do this by associating a quasi-order to each compatible C-relation of a group, and then give the structure of such quasi-ordered groups. We also reformulate in terms of quasi-orders some results concerning C-minimal groups given in [5].

We exhibit infinitely many hyperbolic 3-manifold groups that are not right-orderable.

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of ${{S}^{3}}$ branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of ${{S}^{3}}$ branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the ${{5}_{2}}$ knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be $\ell$-embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups.

THEOREM. Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be$\ell$-embedded in a finitely presented lattice-ordered group.

If $\xi$ is a real number, let $D(\xi)$ be the Abelian rank 2 group $\Z^2$ with order $(m,n)>0$ if and only if $m+n\xi>0$.

COROLLARY. $D(\xi)$can be$\ell$-embedded in a finitely presented lattice-ordered group if and only if$\xi$is a recursive real number.

Thus an algebraic characterisation of recursive real numbers is obtained. In particular, $\pi$ is ‘$\ell$-algebraic’ in that it can be captured by finitely many relations in this language.

A proof is given of the following theorem, which characterizes full automorphism groups of ordered abelian groups: a group $H$ is the automorphism group of some ordered abelian group if and only if $H$ is right-orderable.

An additive subgroup P of a skew field F is called a prime of F if P does not contain the identity, but if the product xy of two elements x and y in F is contained in P, then x or y is in P. A prime segment of F is given by two neighbouring primes P 1 ⊃ P 2; such a segment is invariant, simple, or exceptional depending on whether A(P 1) = {a ∈ P 1 | P 1 aP 1 ⊂ P 1} equalsP 1, P 2 or lies properly between P 1 and P 2. The set T(F) of all primes of F together with the containment relation is a tree if |T(F)| is finite, and 1 < |T(F)| < ∞ is possible if F is not commutative. In this paper we construct skew fields with prescribed types of sequences of prime segments as skew fields F of fractions of group rings of certain right ordered groups. In particular, groups G of affine transformations on ordered vector spaces V are considered, and the relationship between properties of Dedekind cuts of V, certain right orders on G, and chains of prime segments of F is investigated. A general result in Section 4 describing the possible orders on vector spaces over ordered fields may be of independent interest.

The object of this paper is to show that every soluble right orderable group is locally indicable. The proof identifies an interesting connection between the theory of right orderable groups and the theory of amenable groups and bounded cohomology.

We extend to amenable right ordered groups the theorems of Nehari and Hartman on Hankel matrices.