Isolated orderings on amalgamated free products

We show that an amalgamated free product $G*_{A}H$ admits a discrete isolated ordering, under some assumptions of $G,H$ and $A$. This generalizes the author's previous construction of isolated orderings, and unlike known constructions of isolated orderings, can produce an isolated ordering with many non-trivial proper convex subgroups.Comment: 10 pages, 3 figure

Π10 classes and orderable groups

AbstractIt is known that the spaces of orders on orderable computable fields can represent all Π10 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Π10 classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes

Ordering the braid groups

We give an explicit geometric argument that Artin's braid group $B_n$ is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical form. The left-consistent form of a braid which is positive (respectively negative) in our order has consistently positive (respectively negative) exponent in the smallest braid generator which occurs. It follows that our ordering is identical to that of Dehornoy, constructed by very different means, and we recover Dehornoy's main theorem that any braid can be put into such a form using either positive or negative exponent in the smallest generator but not both. Our definition of order is strongly connected with Mosher's normal form and this leads to an algorithm to decide whether a given braid is positive, trivial, or negative which is quadratic in the length of the braid word.Comment: 24 pages, 10 figure

Orders On Free Metabelian Groups

A bi-order on a group $G$ is a total, bi-multiplication invariant order. Such an order is regular if the positive cone associated to the order can be recognised by a regular language. A subset $S$ in an orderable group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to $S$. In this paper, we study the convex hull of the derived subgroup of a free metabelian group with respect to a bi-order. As an application, we prove that non-abelian free metabelian groups of finite rank do not admit a regular bi-order while they are computably bi-orderable.Comment: 19 Pages, 1 figure. Comments are welcome

Regular left-orders on groups

A regular left-order on finitely generated group $G$ is a total, left-multiplication invariant order on $G$ whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and give a classification of the groups all whose left-orders are regular left-orders. In addition, we prove that solvable Baumslag-Solitar groups $B(1,n)$ admits a regular left-order if and only if $n\geq -1$. Finally, Hermiller and Sunic showed that no free product admits a regular left-order, however we show that if $A$ and $B$ are groups with regular left-orders, then $(A*B)\times \mathbb{Z}$ admits a regular left-order.Comment: 41 pages,9 figure

Computability Theory and Ordered Groups

Ordered abelian groups are studied from the viewpoint of computability theory. In particular, we examine the possible complexity of orders on a computable abelian group. The space of orders on such a group may be represented in a natural way as the set of infinite paths through a computable tree, but not all such sets can occur in this way. We describe the connection between the complexity of a basis for a group and an order for the group, and completely characterize the degree spectra of the set of bases for a group. We describe some restrictions on the possible degree spectra of the space of orders, including a connection to algorithmic randomness