[Paper Review] The quadratic isoperimetric inequality for mapping tori of free group automorphisms I: Positive automorphisms

Martin R. Bridson, Daniel Groves|ArXiv.org|Nov 29, 2002

Geometric and Algebraic Topology18 references3 citations

TL;DR

This paper establishes that the mapping torus of a positive automorphism of a finitely generated free group satisfies a quadratic isoperimetric inequality, proving that its Dehn function is at most quadratic. The authors use a geometric analysis of van Kampen diagrams, focusing on corridor subdiagrams and their time-evolving structure under the automorphism, to bound the area of such diagrams in terms of the boundary length, thereby proving the quadratic bound without relying on automaticity or non-positive curvature.

ABSTRACT

If $F$ is a finitely generated free group and $ϕ$ is a positive automorphism of $F$ then $F times_ϕZ$ satisfies a quadratic isoperimetric inequality.

Motivation & Objective

• To establish a quadratic isoperimetric inequality for the mapping torus $F\rtimes_{\phi}\mathbb{Z}$ when $\phi$ is a positive automorphism of a finitely generated free group $F$.
• To overcome the limitations of prior methods—such as automaticity or non-positive curvature—by developing a new geometric approach to Dehn function bounds.
• To analyze the large-scale geometry of van Kampen diagrams, particularly the evolution of corridor subdiagrams under the automorphism's time flow.
• To provide a foundation for extending the result to all free group automorphisms using relative train track technology in subsequent work.

Proposed method

• Analyzes van Kampen diagrams over natural presentations of free-by-cyclic groups, focusing on corridor subdiagrams and their time-evolving structure.
• Introduces the concept of 'preferred future' to trace the trajectory of 1-cells in the corridor flow and control their cancellation behavior.
• Uses bounded cancellation constants and time-based estimates to bound the length of swollen futures of corridor components.
• Applies a decomposition of corridors into colored regions and teams to control area contributions via combinatorial and geometric bounds.
• Employs lemmas on neutering, pincer effects, and 2-coloring to manage cancellation patterns and limit the growth of uncancelled edges.
• Derives explicit constants $K_1$ and $K$ in terms of $B$, $M$, $T_0$, $T_1$, $C_0$, $C_1$, and $\lambda_0$ to bound the total area by $K|\partial\Delta|^2$.

Experimental results

Research questions

• RQ1 Does the mapping torus of a positive automorphism of a finitely generated free group satisfy a quadratic isoperimetric inequality?
• RQ2 Can the Dehn function of such groups be bounded quadratically without relying on automaticity or non-positive curvature?
• RQ3 How do corridor subdiagrams evolve over time in van Kampen diagrams, and can their area be controlled via geometric and combinatorial estimates?
• RQ4 What role do bounded cancellation, pincer effects, and neutering behavior play in controlling the growth of uncancelled edges in the diagram?
• RQ5 Can the structure of the automorphism's action be leveraged to bound the total area of a van Kampen diagram in terms of its boundary length?

Key findings

• The mapping torus $F\rtimes_{\phi}\mathbb{Z}$ satisfies a quadratic isoperimetric inequality when $\phi$ is a positive automorphism of a finitely generated free group $F$.
• The area of any van Kampen diagram $\Delta$ is bounded by $\frac{K}{2}|\partial\Delta|^2$, where $K = 2C_0 + 2K_1 + 2B + 1$ is an explicit constant depending on the automorphism's parameters.
• The proof establishes that the total length of all corridors in a diagram is bounded linearly in $|\partial\Delta|$, with the constant $K$ explicitly constructed from $B$, $M$, $T_0$, $T_1$, $C_0$, $C_1$, and $\lambda_0$.
• The analysis shows that the swollen future of any corridor component shrinks by at most $2B$ per time step, and its length is bounded by $2B|\text{up}(\mathcal{B})| + |\text{bdy}(\mathcal{B})|$, leading to a global linear bound on area.
• The method successfully avoids reliance on automaticity or non-positive curvature, providing a new geometric framework applicable to non-automatic and non-negatively curved groups.
• The result extends verbatim to automorphisms with a train track representative, and the full result for all automorphisms is established in a sequel using relative train track technology.