[Paper Review] Energy Games over Totally Ordered Groups

Kozachinskiy, Alexander|arXiv (Cornell University)|Aug 25, 2014

Geometric and Algebraic Topology127 references52 citations

TL;DR

This paper establishes that the Thompson-like group T(φ) associated with a minimal subshift (X, φ) is finitely-generated, left-orderable, and simple. By constructing a suspension group from a minimal subshift using dyadic flow boxes and leveraging group actions on Cantor sets, the authors prove that the group's structure—generated by a base group and specific flow-box subgroups—yields a simple, left-orderable, finitely generated group. The key contribution is a general construction of such groups from symbolic dynamics.

ABSTRACT

Kopczyński (ICALP 2006) conjectured that prefix-independent half-positional winning conditions are closed under finite unions. We refute this conjecture over finite arenas. For that, we introduce a new class of prefix-independent bi-positional winning conditions called energy conditions over totally ordered groups. We give an example of two such conditions whose union is not half-positional. We also conjecture that every prefix-independent bi-positional winning condition coincides with some energy condition over a totally ordered group on periodic sequences.

Motivation & Objective

• To construct a new class of finitely-generated, left-orderable, simple groups using symbolic dynamics.
• To establish that the suspension group T(φ) associated with a minimal subshift is simple and finitely generated.
• To demonstrate that the group T(φ) admits a left-invariant total order.
• To provide a general framework linking dynamical systems and orderable group theory.

Proposed method

• Define the suspension group T(φ) as an extension of the group of homeomorphisms on a Cantor set via dyadic flow boxes.
• Use a generating partition {C₁, ..., C_d} of the minimal subshift (X, φ) to define flow-box subgroups FC_i,J.
• Apply the equivariance property and conjugation by eT to propagate generators across all dyadic intervals of length < 1.
• Leverage the simplicity of F′_J (Thompson's group F) and group commutator identities to show closure under intersections and unions of clopen sets.
• Use Lemma 4.5.15 to prove that subgroups F′_C,J are contained in the subgroup generated by FC_i,J and eT.
• Conclude that T(φ) is finitely generated by showing it is generated by eT and finitely many FC_i,I subgroups.

Experimental results

Research questions

• RQ1 Can a minimal subshift be used to construct a finitely-generated, left-orderable, simple group?
• RQ2 What conditions ensure that the suspension group T(φ) is simple and finitely generated?
• RQ3 How does the structure of the generating partition {C₁, ..., C_d} affect the group T(φ)?
• RQ4 What role do dyadic flow boxes and their associated subgroups play in the group's generation?
• RQ5 Is the group T(φ) left-orderable when (X, φ) is a minimal subshift?

Key findings

• The group T(φ) is finitely-generated when (X, φ) is a minimal subshift with a finite generating partition.
• T(φ) is simple, as shown by proving that every normal subgroup must be the whole group via closure under group operations.
• T(φ) admits a left-invariant total order, making it a left-orderable group.
• The subgroup generated by eT and FC_i,I for i ∈ {1, ..., d} equals T(φ), confirming finite generation.
• The proof relies on the fact that F′_J is simple and that commutators generate the group, enabling closure under set operations.
• The construction generalizes to any subshift, not just minimal ones, though minimality ensures simplicity and the desired group structure.