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Linear isoperimetric characterisation of hyperbolic groups
Statement
A finitely generated group is hyperbolic if and only if it admits a finite presentation satisfying a linear isoperimetric inequality for van Kampen area: there is a constant such that every null-homotopic word has a van Kampen diagram with at most -cells.
Facts & Assumptions
Given: A finitely generated group .
A finite Dehn presentation gives a linear isoperimetric inequality by shortening a trivial word one relator cell at a time.
A finite presentation with linear isoperimetric inequality yields uniformly thin geodesic bigons, and hence a hyperbolic Cayley graph.
Hyperbolic groups admit finite Dehn presentations (Hyperbolic groups admit finite Dehn presentations).
Proof
If is hyperbolic, then [L1] supplies a finite Dehn presentation, and [A1] turns that Dehn property into a linear isoperimetric inequality.
Conversely, if a finite presentation satisfies a linear isoperimetric inequality, then [A2] gives thin geodesic bigons and hence a hyperbolic Cayley graph. Therefore the two conditions are equivalent.
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Sources
- Clara Löh, Geometric Group Theory, Section 6.4 (standard reference, not scraped)
- Brian H. Bowditch, A course on geometric group theory, Section 2.3 (standard reference, not scraped)