[Paper Review] Isolated points in the space of left orderings of a group

Adam Clay|ArXiv.org|Dec 12, 2008

Finite Group Theory Research5 references3 citations

TL;DR

This paper investigates isolated points in the space of left orderings of a group, extending prior results to uncountable groups. It proves that no dense left ordering is isolated, and that the closure of dense orderings forms a dense $G_{\delta}$ subset homeomorphic to a Cantor set. For braid groups, the Dehornoy ordering is shown to be an accumulation point of its conjugates, not isolated, despite being discrete.

ABSTRACT

Let G be a left orderable group and LO(G) the space of all left orderings. We investigate the circumstances under which a left ordering < of G can correspond to an isolated point in LO(G), in particular we extend known results to cover the case of uncountable groups. With minor technical restrictions on the group G, we also find that no dense left ordering is isolated in LO(G), and that the closure of the set of all dense left orderings of G yields a dense G-delta set within a Cantor set of left orderings in LO(G). Lastly, we show that certain conditions on a discrete left ordering of G can guarantee that it is not isolated in LO(G), and we illustrate these ideas using the Dehornoy ordering of the braid groups.

Motivation & Objective

• To determine conditions under which a left ordering of a group corresponds to an isolated point in the space $LO(G)$, especially for uncountable groups.
• To generalize the main result of [9] on isolated left orderings to uncountable groups, under mild technical restrictions.
• To analyze the topological structure of $LO(G)$, particularly the role of dense and discrete orderings.
• To investigate whether the Dehornoy ordering of braid groups is isolated, using conjugacy dynamics and convex subgroup structure.
• To characterize when a discrete left ordering fails to be isolated, based on its convex subgroup structure.

Proposed method

• Uses the positive cone representation of left orderings, identifying $LO(G)$ as a closed subspace of $2^G$ with the product topology.
• Applies the $G$-action on $LO(G)$ via conjugation of positive cones, analyzing orbit accumulation points.
• Employs the Conradian soul $C_< (G)$ and bi-ordering convex subgroup $B_< (G)$ as invariants to detect isolation.
• Applies the subword property (property S) of the Dehornoy ordering: $\beta \alpha \beta^{-1} \in P_D$ for $\alpha \in B_n^+$, $\beta \in B_n$.
• Uses the shift homomorphism $sh: B_m \to B_n$ ($m < n$) to identify convex subgroups as shifted braid groups.
• Proves that all convex subgroups in the Dehornoy ordering are isomorphic to $sh^r(B_{n-r})$, and none are bi-orderable except $\langle \sigma_{n-1} \rangle$.

Experimental results

Research questions

• RQ1 Under what conditions is a left ordering of a group isolated in $LO(G)$, especially for uncountable groups?
• RQ2 Can the Dehornoy ordering of the braid group $B_n$ be an isolated point in $LO(B_n)$?
• RQ3 What topological structure does the set of dense left orderings of a countable group $G$ possess?
• RQ4 When is a discrete left ordering not isolated, and how does its convex subgroup structure affect this?
• RQ5 Does the closure of the set of dense orderings in $LO(G)$ form a Cantor set, and under what conditions?

Key findings

• No dense left ordering of a group $G$ is isolated in $LO(G)$, under mild technical conditions.
• The closure of the set of dense left orderings in $LO(G)$ is a dense $G_{\delta}$ subset homeomorphic to the Cantor set.
• For the braid group $B_n$ with $n > 2$, the Dehornoy ordering $P_D$ is not isolated, as it is an accumulation point of its conjugates.
• The Dehornoy ordering of $B_n$ is discrete, with smallest positive element $\sigma_{n-1}$, and satisfies the subword property.
• All convex subgroups in the Dehornoy ordering of $B_n$ are isomorphic to shifted braid groups $sh^r(B_{n-r})$, and none are bi-orderable except $\langle \sigma_{n-1} \rangle$.
• Since $P_D$ has no nontrivial bi-orderable convex subgroups properly containing $\langle \sigma_{n-1} \rangle$, it is not isolated, as per Theorem 4.9.