[Paper Review] The quadratic isoperimetric inequality for mapping tori of free group automorphisms II: The general case

Martin R. Bridson, Daniel Groves|ArXiv.org|Oct 10, 2006

Geometric and Algebraic Topology10 references4 citations

TL;DR

This paper establishes a quadratic isoperimetric inequality for mapping tori of automorphisms of finitely generated free groups, extending previous results from positive automorphisms to the general case using improved relative train track maps and a novel concept of 'beads'. The key contribution is a linear bound on the number of beads along the bottom of corridors in van Kampen diagrams, which, combined with geometric and algorithmic arguments, implies that the word problem in such groups is solvable in non-deterministic quadratic time.

ABSTRACT

If F is a finitely generated free group and ϕis an automorphism of F then the mapping torus of ϕadmits a quadratic isoperimetric inequality. This is the third and final paper in a series proving this theorem. The first two were math.GR/0211459 and math.GR/0507589.

Motivation & Objective

• To prove that the mapping torus of any automorphism of a finitely generated free group satisfies a quadratic isoperimetric inequality.
• To extend the proof strategy from positive automorphisms to the general case by introducing and analyzing 'beads' as structural units in van Kampen diagrams.
• To establish a linear bound on the number of beads along the bottom of corridors in van Kampen diagrams, which is central to the isoperimetric inequality.
• To deduce that the conjugacy problem is solvable in such groups, building on prior results in geometric group theory.
• To provide a non-deterministic quadratic time algorithm for the word problem in free-by-cyclic groups using bracketing and geometric bounds.

Proposed method

• Adapt the proof strategy from the positive automorphism case using improved relative train track maps and the theory of 'beads' as the fundamental building blocks.
• Define 'beads' as the generalization of single edges (letters) in the positive case, and prove their existence and structural control via the Beaded Decomposition Theorem.
• Use the Pincer Lemma and team-length bounds to control the geometry of van Kampen diagrams, particularly in stack diagrams where singularities are absent.
• Implement a 'Bonus Scheme' to account for geometric inefficiencies, with modifications to handle the absence of t-edges in stack diagrams.
• Apply the structure of stack diagrams (simple sequences of corridors) to eliminate dependencies on t-edges in key bounds, simplifying the analysis.
• Reduce the isoperimetric problem to bounding the number of beads in corridors, leveraging the fact that bead length is not globally bounded but their count is.

Experimental results

Research questions

• RQ1 Can the quadratic isoperimetric inequality be established for mapping tori of arbitrary automorphisms of finitely generated free groups, not just positive ones?
• RQ2 How can the concept of 'beads' serve as a structural replacement for single edges in the context of general automorphisms?
• RQ3 What modifications are required in the proof framework of the positive case to handle non-positive automorphisms and their associated train track maps?
• RQ4 How can the geometric control of van Kampen diagrams be maintained in the absence of global bounds on bead length?
• RQ5 Can the word problem in free-by-cyclic groups be solved in non-deterministic quadratic time using the derived geometric bounds?

Key findings

• The mapping torus $F times_ ho bZ$ of any finitely generated free group $F$ and automorphism $ ho$ satisfies a quadratic isoperimetric inequality.
• A linear bound exists on the number of beads along the bottom of any corridor in a van Kampen diagram over the mapping torus, which is the central geometric estimate.
• The conjugacy problem for $F times_ ho bZ$ is solvable, as a consequence of the Dehn function being bounded by $n^2$.
• The word problem in such groups admits a non-deterministic quadratic time algorithm, with the bound expressed via $t$-complete bracketings and a constant $K = K( ho, bB)$.
• In stack diagrams (a special class of van Kampen diagrams), the contribution of $t$-edges to geometric bounds can be eliminated, simplifying the analysis and allowing direct application of the Pincer Lemma.
• The proof reduces to verifying that the number of beads is linear in $|d riangle|$, and that the bonus scheme and team-length bounds remain effective despite the absence of $t$-edge dependencies in stack diagrams.