Everything about Dehn function
Group theory function
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group in terms of the length of that relation. The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive. The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Dehn functionA 5-minute orientation
- Skim a recent paper: Weak commutativity, virtually nilpotent groups, and Dehn functionsInfluential work since 2022
- Pick up a classic book: Dickens studies annualThe most-reprinted book on the subject
- Read the landmark paper: The Dehn function of Richard Thompson’s group F is quadraticCited 21 times
- Go sideways: Van Kampen diagramA closely related topic
- Go sideways: Word problem for groupsA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
The Dehn function of Richard Thompson’s group F is quadratic
The Dehn function of SL(n;Z)
Metabelian groups with quadratic Dehn function and Baumslag–Solitar groups
Nilpotent groups without exactly polynomial Dehn function
Subquadratic Growth of the Averaged Dehn Function for Free Abelian Groups
On quadratic Dehn functions
Dehn function and asymptotic cones of Abels’ group
The Dehn Function of Stallings’ Group
🆕 Influential since 2022
📚 Books worth reading
Dickens studies annualStanley Friedman, Edward Guiliano · 2006
Sherlock Holmes in BabylonMarlow Anderson, Victor J. Katz · 2004
Die Legendre'schen Sätze über die Winkelsumme in DreieckMax Dehn · 1900- 📕Torsion Subgroups of Groups with Quadratic Dehn FunctionFrancis Wagner · 2025
Topology in molecular biologyMikhail Ilʹich Monastyrskiĭ · 2007
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
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