On spaces of Conradian group orderings

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 space of $C$-orderings. As a relevant example, we carefully study the case of Baumslag-Solitar's group B(1,2). We show that B(1,2) has four $C$-orderings, each of which is bi-invariant, but its space of left-orderings is homeomorphic to the Cantor set

Left-orderings on free products of groups

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 techniques, we also give a new and constructive proof of the fact that the natural conjugation action of the free group (on two or more generators) on its space of left-orderings has a dense orbit.Comment: 13 page

Describing all bi-orderings on Thompson's group F

We describe all possible ways of bi-ordering Thompson group F: its space of bi-orderings is made up of eight isolated points and four canonical copies of the Cantor set.Comment: Final versio

The space of relative orders and a generalization of Morris indicability theorem

Electronic version of an article published as Journal of Topology and Analysis 13.1 (2021): 75-85. https://doi.org/10.1142/S1793525320500053 © 2021 World Scientific Publishing Company. https://www.worldscientific.com/worldscinet/jtaWe introduce the space of relative orders on a group and show that it is compact whenever the group is finitely generated. We use this to show that if G is a finitely generated group acting on the line by order preserving homeomorphisms and some stabilizer of a point is a proper and co-amenable subgroup, then G surjects onto Z. This is a generalization of a theorem of MorrisThe first author is supported by the Juan de la Cierva grant IJCI-2014-22425, and acknowledges partial support from the Spanish Government through grants number MTM2014-54896 and MTM2017-82690-P and through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). The second author is partially supported by FONDECYT 118154