[Paper Review] Construction of isolated left orderings via partially central cyclic amalgamation

Tetsuya Ito|arXiv (Cornell University)|Jul 4, 2011

Finite Group Theory Research7 references3 citations

TL;DR

This paper introduces a novel construction method—partially central cyclic amalgamation—for generating isolated left orderings in groups, particularly producing centerless groups with finitely generated positive cones. By amalgamating groups with known isolated orderings using a non-central, cofinal element as a connector, the method yields new examples, including groups with $2(m-1)!$ distinct $G$-orbits of isolated orderings and $(m-1)!$ distinct $ extrm{Aut}(G)$-orbits when parameters are distinct.

ABSTRACT

We give a new method to construct isolated left orderings of groups whose positive cones are finitely generated. Our construction uses an amalgamated free product of two groups having an isolated ordering. We construct a lot of new examples of isolated orderings, and give an example of isolated left orderings having various properties which previously known isolated orderings do not have.

Motivation & Objective

• To develop a general construction method for isolated left orderings in groups, especially those with finitely generated positive cones.
• To address the scarcity of known examples of genuine isolated left orderings, particularly in centerless groups.
• To extend existing constructions by incorporating non-central, cofinal elements in amalgamated free products.
• To explore the orbit structure of isolated orderings under group and automorphism actions, quantifying their diversity.

Proposed method

• Utilizes the partially central cyclic amalgamation of two groups $G_1$ and $G_2$, each equipped with an isolated left ordering.
• Selects a non-central, $<_{G}$-cofinal element $z_H$ in one factor group $H$ to serve as the amalgamating element.
• Applies Theorem 1.1 to construct a new group $G = G_1 *_H G_2$ with an isolated left ordering whose positive cone is finitely generated.
• Employs the right invariance of the ordering under multiplication by $z_H$ and $z_H^k$ to ensure the positivity of the resulting cone.
• Re-expresses the resulting group in terms of new generators to reveal structural properties such as trivial center.
• Analyzes the action of $G$ and $ extrm{Aut}(G)$ on $ extrm{LO}(G)$ to count distinct orbits of isolated orderings via minimal positive elements.

Experimental results

Research questions

• RQ1 Can isolated left orderings be systematically constructed in centerless groups using amalgamated free products?
• RQ2 How many distinct $G$-orbits of isolated orderings can arise from a single construction method?
• RQ3 What role does the choice of non-central, cofinal element play in ensuring the isolatedness of the resulting ordering?
• RQ4 Can the automorphism group action reveal additional diversity in isolated orderings beyond group action orbits?
• RQ5 Under what conditions does the amalgamation process preserve the isolatedness and finite generation of the positive cone?

Key findings

• The construction yields a centerless group $H_{p,q,m,n} = igracevert a,b,c igracevert b^m = c^n, a^p = (bc)^q \big\}$ with an isolated left ordering.
• The isolated left ordering of $H_{p,q,m,n}$ has a positive cone generated by $\{a(bc)^{1-q}, bc^{1-n}, c\}$, confirming finite generation.
• For the group $G$ formed from $m$ infinite cyclic groups amalgamated over a common element, at least $2(m-1)!$ distinct $G$-orbits of isolated orderings exist.
• When all $a_i$ are distinct, the automorphism group $\textrm{Aut}(G)$ acts on $\textrm{LO}(G)$ with at least $(m-1)!$ distinct orbits derived from isolated orderings.
• The method generalizes known examples: the Dubrovina-Dubrovin ordering and the $\mathbb{Z}*_{\mathbb{Z}}\mathbb{Z}$ orderings are recovered as special cases.
• The construction demonstrates that isolated orderings can exist in groups with trivial center, resolving a gap in the known landscape of such groups.