8.3 Examples and classification results

Dehn functions measure how hard it is to solve the word problem in groups. They're like a speedometer for group theory, showing how fast area grows in diagrams that prove words equal the identity.

Examples range from linear functions in hyperbolic groups to quadratic in abelian ones. Some groups even have exponential or non-elementary recursive Dehn functions, revealing the rich landscape of group behavior.

Dehn Functions: Types and Examples

Linear and Quadratic Dehn Functions

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• Dehn function measures area of minimal-area diagrams for words representing identity in group's presentation
• Linear Dehn functions characterize hyperbolic groups (free groups, fundamental groups of closed hyperbolic manifolds)
• Quadratic Dehn functions appear in groups like $Z^2$ (free abelian group on two generators) and fundamental group of torus
• Constructing explicit van Kampen diagrams analyzes area growth for Dehn functions
• Examples of linear Dehn function groups include $F_2$ (free group on two generators) and $PSL(2, Z)$ (modular group)
• Quadratic Dehn function examples encompass $Z \times Z$ (direct product of two infinite cyclic groups) and $\pi_1(T^2)$ (fundamental group of 2-torus)

Higher Degree and Non-Elementary Recursive Dehn Functions

• Exponential Dehn functions occur in Baumslag-Solitar groups $BS(1,2)$ and certain solvable groups
• Polynomial Dehn functions of higher degree found in nilpotent groups and wreath products
  • Cubic Dehn function example $H_3(Z)$ (3-dimensional Heisenberg group)
  • Quartic Dehn function example $F_2 \wr Z$ (wreath product of free group with integers)
• Some groups have Dehn functions not elementary recursive (Thompson's group F)
• Non-elementary recursive Dehn function example $SL(n,Z)$ for $n \geq 5$ (special linear group over integers)

Group Classification by Growth Rate

Isoperimetric Spectrum and Function Classes

• Groups classified into isoperimetric spectrum classes based on asymptotic behavior of Dehn functions
• Isoperimetric spectrum forms partially ordered set of equivalence classes under asymptotic equivalence
• Hyperbolic groups characterized by linear Dehn functions, lowest non-trivial class in isoperimetric spectrum
• CAT(0) groups and automatic groups have at most quadratic Dehn functions
• Finitely presented groups always have at most exponential Dehn functions
• Groups with intermediate growth Dehn functions exist ($n^\alpha$ for non-integer $\alpha > 2$)
• Example of intermediate growth $SL(2,Z[1/p])$ (special linear group over $Z[1/p]$) with Dehn function $n^{2 \log_2(3)}$

Classification Challenges and Open Questions

• Classification based on Dehn functions remains incomplete
• Open questions persist about functions realizable as Dehn functions of groups
• Difficulty in determining exact Dehn functions for many groups
• Example $Out(F_n)$ (outer automorphism group of free group) with unknown exact Dehn function
• Ongoing research explores Dehn functions of specific group classes (one-relator groups, 3-manifold groups)

Geometry and Dehn Functions

Geometric Interpretations and Connections

• Dehn function intimately connected to geometry of group's Cayley graph
• Hyperbolicity of Cayley graph equivalent to linear Dehn function
• Quadratic Dehn functions often relate to non-positively curved spaces
• Growth rate of Dehn function reflects difficulty of solving word problem in group
• Asymptotic cones of groups provide geometric insight into Dehn function behavior
• Filling function in Riemannian geometry analogous to Dehn function in group theory
• Quasi-isometries between groups preserve asymptotic behavior of Dehn functions, enabling geometric classification

Examples of Geometric Connections

• Hyperbolic groups ($F_2$, $PSL(2,Z)$) have linear Dehn functions due to negative curvature properties
• CAT(0) groups ($Z^n$, right-angled Artin groups) have at most quadratic Dehn functions
• Nilpotent groups (Heisenberg group) exhibit polynomial Dehn functions related to their graded structure
• Solvable groups ($BS(1,2)$) often have exponential Dehn functions reflecting their exponential distortion

Open Problems in Dehn Function Classification

Conjectures and Unresolved Questions

• Dehn function gap conjecture posits no groups with Dehn functions strictly between $n \log n$ and $n^2$
• Unknown existence of groups with Dehn functions $n^\alpha$ for non-integer $\alpha$ between 2 and 3
• Relationship between Dehn functions and other group invariants (growth functions, cohomological dimension) actively researched
• Question of functions realizable as Dehn functions of finitely presented groups remains open
• Computational complexity of determining Dehn function for given finitely presented group not fully understood

Interdisciplinary Connections and Future Directions

• Connections between Dehn functions and other mathematics areas explored
  • Logic connections (decidability of word problem related to Dehn function growth)
  • Complexity theory links (time complexity of word problem algorithms)
• Ongoing investigations into Dehn functions of specific group classes
  • One-relator groups (Magnus-Moldavanskii hierarchy)
  • 3-manifold groups (geometrization theorem implications)
• Potential applications of Dehn function theory to other fields
  • Cryptography (group-based cryptosystems)
  • Theoretical computer science (complexity classes defined by group properties)