Regular left-orders on groups

Abstract

A regular left-order on finitely generated group a 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 ≥ −1. Finally, Hermiller and Sunić 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) × Z admits a regular left-order.

On the geometry of positive cones in finitely generated groups

“…We use a similar but more involved argument in Theorem 3. 4, where we show that fundamental groups of hyperbolic surfaces do not have coarsely connected positive cones. In this case, 𝐺 acts properly and co-compactly on its Cayley graph, a space which is quasi-isometric to the hyperbolic plane ℍ 2 .…”

“…For instance, regularity passes to finite index subgroups [32] while finite generation does not. † We recommend [4] for an introduction. It is easy to show that finitely presented groups with a regular positive cone have solvable word problem (see [4]).…”

“…† We recommend [4] for an introduction. It is easy to show that finitely presented groups with a regular positive cone have solvable word problem (see [4]). A more refined criterium for not admitting regular positive cones was obtained by Hermiller and S ȗnić in [17] where they show that no positive cone in a free product of groups is regular.…”

“…Another family of examples of groups with the Prieto property are left-orderable groups for which every order can be described by a regular language. This family is described in detail in [4].…”

“…First we construct the fundamental domain 𝑇 0 and the generating set 𝑋. After that we shall verify the properties (1)- (4). □…”