Spaces of left-preorders on free products
Iv\'an Ch\'ercoles Cuesta
TL;DR
This paper investigates the topological structure of the space of left-preorders on free products of groups, showing it is either empty or a Cantor set when groups are finitely generated, with extensions to subgroups of finite Kurosh rank.
Contribution
It introduces new dynamical techniques to analyze the topology of left-preorder spaces on free products and their subgroups, revealing their Cantor set structure.
Findings
No isolated elements in the space of left-preorders on free products.
The space is either empty or a Cantor set for finitely generated groups.
Provides a generating set for intersections of certain subgroups.
Abstract
Using dynamical techniques we show that there are no isolated elements on the space of left-preorders on a free product of two groups. As a consequence, when the groups are finitely generated, this space is either empty or a Cantor set. For any subgroup of a free product having finite Kurosh rank, we develop similar results for the subspace consisting on the set of left-preorders relative to that subgroup. We provide a generating set for a certain intersection of subgroups of a free product.
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Taxonomy
TopicsAdvanced Topology and Set Theory